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URL: https://oeis.org/A257622

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A257622
Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.
8
1, 4, 4, 16, 56, 16, 64, 552, 552, 64, 256, 4696, 11040, 4696, 256, 1024, 36968, 171448, 171448, 36968, 1024, 4096, 278232, 2305968, 4457648, 2305968, 278232, 4096, 16384, 2037736, 28346088, 94844912, 94844912, 28346088, 2037736, 16384
OFFSET
0,2
FORMULA
T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.
Sum_{k=0..n} T(n, k) = A051605(n).
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 3, and b = 4. - G. C. Greubel, Mar 20 2022
EXAMPLE
Triangle begins as:
1;
4, 4;
16, 56, 16;
64, 552, 552, 64;
256, 4696, 11040, 4696, 256;
1024, 36968, 171448, 171448, 36968, 1024;
4096, 278232, 2305968, 4457648, 2305968, 278232, 4096;
16384, 2037736, 28346088, 94844912, 94844912, 28346088, 2037736, 16384;
MATHEMATICA
T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n, k, 3, 4], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
PROG
(SageMath)
def T(n, k, a, b): # A257622
if (k<0 or k>n): return 0
elif (n==0): return 1
else: return (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b)
flatten([[T(n, k, 3, 4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022
CROSSREFS
See similar sequences listed in A256890.
Sequence in context: A222144 A342817 A107382 * A185567 A203536 A375990
KEYWORD
nonn,tabl
AUTHOR
Dale Gerdemann, May 10 2015
STATUS
approved