Template:Seal integer sequences/formulas

One

Oak

A022166

data
🌊    triangle Oak (A022166)    row sums Daisy (A006116) k n 0 1 2 3 4 5 6 7 sums 0 1 1 1 1 1 2 2 1 3 1 5 3 1 7 7 1 16 4 1 15 35 15 1 67 5 1 31 155 155 31 1 374 6 1 63 651 1395 651 63 1 2825 7 1 127 2667 11811 11811 2667 127 1 29212

$\mathrm {Oak} (n,k)~~=~~\mathrm {Oak} (n-1,k-1)~+~2^{k}\cdot \mathrm {Oak} (n-1,k)$
${\begin{matrix}\mathrm {Oak} (n,k)&~~=~~&\mathrm {Oak} (n-1,k-1)&~+~&2^{k}&\cdot &\mathrm {Oak} (n-1,k)\\\\\mathrm {Oak} (5,3)&~~=~~&\mathrm {Oak} (4,2)&~+~&2^{3}&\cdot &\mathrm {Oak} (4,3)\\155&~~=~~&35&~+~&8&\cdot &15\end{matrix}}$

Birch

A139382 with trivial column on the left

data
triangle Birch (~A139382)    row sums Aster (A135922) k n 0 1 2 3 4 5 6 7 sums 0 1 1 1 0 1 1 2 0 1 1 2 3 0 1 4 1 6 4 0 1 13 11 1 26 5 0 1 40 90 26 1 158 6 0 1 121 670 480 57 1 1330 7 0 1 364 4811 7870 2247 120 1 15414

$\mathrm {Birch} (n,k)~~=~~\mathrm {Birch} (n-1,k-1)~+~(2^{k}-1)\cdot \mathrm {Birch} (n-1,k)$
${\begin{matrix}\mathrm {Birch} (n,k)&~~=~~&\mathrm {Birch} (n-1,k-1)&~+~&(2^{k}-1)&\cdot &\mathrm {Birch} (n-1,k)\\\\\mathrm {Birch} (5,3)&~~=~~&\mathrm {Birch} (4,2)&~+~&(2^{3}-1)&\cdot &\mathrm {Birch} (4,3)\\90&~~=~~&13&~+~&7&\cdot &11\end{matrix}}$

Two

Birch and Oak

$\mathrm {Oak} (n,k)~~=~~\sum _{i=0}^{k}\mathrm {Birch} (n-i,n-k)\cdot {\binom {n}{i}}$

${\begin{matrix}\mathrm {Oak} (n,k)&~~=~~&\sum _{i=0}^{k}\mathrm {Birch} (n-i,n-k)\cdot {\binom {n}{i}}\\\\\mathrm {Oak} (5,3)&~~=~~&\sum _{i=0}^{3}\mathrm {Birch} (5-i,5-3)\cdot {\binom {5}{i}}\\155&~~=~~&\mathrm {Birch} (5,2)\cdot {\binom {5}{0}}~+~\mathrm {Birch} (4,2)\cdot {\binom {5}{1}}~+~\mathrm {Birch} (3,2)\cdot {\binom {5}{2}}~+~\mathrm {Birch} (2,2)\cdot {\binom {5}{3}}\\155&~~=~~&40\cdot 1~+~13\cdot 5~+~4\cdot 10~+~1\cdot 10\\\end{matrix}}$

Dahlia and Daisy

Daisy is A006116. Dahlia is A182176 with a leading 1.

data
0 1 2 3 4 5 6 7 Daisy 1 2 5 16 67 374 2825 29212 Dahlia 1 1 3 11 51 307 2451 26387

$\mathrm {Daisy} (n)~~=~~\sum _{i=0}^{n}\mathrm {Dahlia} (i)$
${\begin{matrix}\mathrm {Daisy} (n)&~~=~~&\sum _{i=0}^{n}\mathrm {Dahlia} (i)\\\\\mathrm {Daisy} (5)&~~=~~&\mathrm {Dahlia} (0)~+~...~+~\mathrm {Dahlia} (5)\\374&~~=~~&1+1+3+11+51+307\end{matrix}}$

$\mathrm {Dahlia} (n)~~=~~\mathrm {Daisy} (n-1)~+~(2^{n-1}-1)\cdot \mathrm {Daisy} (n-2)$
${\begin{matrix}\mathrm {Dahlia} (n)&~~=~~&\mathrm {Daisy} (n-1)&~+~&(2^{n-1}-1)&\cdot &\mathrm {Daisy} (n-2)\\\\\mathrm {Dahlia} (6)&~~=~~&\mathrm {Daisy} (5)&~+~&2^{5}-1&\cdot &\mathrm {Daisy} (4)\\2451&~~=~~&374&~+~&31&\cdot &67\end{matrix}}$

Oak and Maple

data

🌊    triangle Oak (A022166)    row sums Daisy (A006116) k n 0 1 2 3 4 5 6 7 sums 0 1 1 1 1 1 2 2 1 3 1 5 3 1 7 7 1 16 4 1 15 35 15 1 67 5 1 31 155 155 31 1 374 6 1 63 651 1395 651 63 1 2825 7 1 127 2667 11811 11811 2667 127 1 29212 Offset Oak with diagonals multiplied by powers of two. 💧    triangle Maple (~A289537)    row sums Dahlia (~A182176) k n 0 1 2 3 4 5 6 7 sums 0   1 1 1   0 1 · 1 1 1 2   0 1 · 2 2 1 · 1 1 3 3   0 1 · 4 4 3 · 2 6 1 · 1 1 11 4   0 1 · 8 8 7 · 4 28 7 · 2 14 1 · 1 1 51 5   0 1 · 16 16 15 · 8 120 35 · 4 140 15 · 2 30 1 · 1 1 307 6   0 1 · 32 32 31 · 16 496 155 · 8 1240 155 · 4 620 31 · 2 62 1 · 1 1 2451 7   0 1 · 64 64 63 · 32 2016 651 · 16 10416 1395 · 8 11160 651 · 4 2604 63 · 2 126 1 · 1 1 26387

Oak is the summation of Maple. E.g. $\mathrm {Oak} (5,3)=\Sigma {\big [}\mathrm {Maple} (3...5,3){\big ]}=1+14+140=155$.

Maple is created by multiplying entries of Oak with powers of two.
For positive $a$ and $d$:   $\mathrm {Maple} (a,d)=\mathrm {Oak} (a-1,d-1)\cdot 2^{a-d}$    E.g. $\mathrm {Maple} (5,3)=\mathrm {Oak} (4,2)\cdot 2^{2}=35\cdot 4=140$