A154228 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1), read by rows.

1, 1, 1, 1, 16, 1, 1, 47, 47, 1, 1, 103, 974, 103, 1, 1, 195, 5354, 5354, 195, 1, 1, 336, 19969, 147068, 19969, 336, 1, 1, 541, 60085, 1259253, 1259253, 60085, 541, 1, 1, 827, 156386, 7010503, 44432886, 7010503, 156386, 827, 1, 1, 1213, 365498, 30299614, 536255794, 536255794, 30299614, 365498, 1213, 1

Offset

0,5

Comments

Row sums are: {1, 2, 18, 96, 1182, 11100, 187680, 2639760, 58768320, ...}.

The row sums of this class of sequences (see cross-references) is given by the following. Let S(n) be the row sum then S(n) = 2*S(n-1) + f(n)*S(n-2) for a given f(n). For this sequence f(n) = (n+1)*(n+2)*(2*n+3)/6 = A000330(n+1). - G. C. Greubel, Mar 02 2021

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.

Example

Triangle begins as:
  1;
  1,    1;
  1,   16,      1;
  1,   47,     47,        1;
  1,  103,    974,      103,         1;
  1,  195,   5354,     5354,       195,         1;
  1,  336,  19969,   147068,     19969,       336,        1;
  1,  541,  60085,  1259253,   1259253,     60085,      541,      1;
  1,  827, 156386,  7010503,  44432886,   7010503,   156386,    827,    1;
  1, 1213, 365498, 30299614, 536255794, 536255794, 30299614, 365498, 1213, 1;

Maple

T:= proc(n, k) option remember;
      if k=0 or k=n then 1
    else T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 02 2021

Mathematica

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + ((n+1)*(n+2)*(2*n+3)/6)*T[n-2, k-1]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *)

Prog

(Sage)
def f(n): return (n+1)*(n+2)*(2*n+3)/6
def T(n, k):
    if (k==0 or k==n): return 1
    else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021
(Magma)
f:= func< n | (n+1)*(n+2)*(2*n+3)/6 >;
function T(n, k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1);
  end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021

Crossrefs

Cf. A154227, A154229, A154230, A154231, A154233.

Cf. A000330.

Sequence in context: A040256 A245910 A133824 * A141697 A202750 A177823

Adjacent sequences: A154225 A154226 A154227 * A154229 A154230 A154231

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 05 2009

Extensions

Edited by G. C. Greubel, Mar 02 2021

Status

approved