A154230 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30)*T(n-2, k-1), read by rows.

1, 1, 1, 1, 100, 1, 1, 455, 455, 1, 1, 1435, 98810, 1435, 1, 1, 3711, 1135370, 1135370, 3711, 1, 1, 8388, 7849141, 464306300, 7849141, 8388, 1, 1, 17161, 40410421, 10431621081, 10431621081, 40410421, 17161, 1, 1, 32495, 169040786, 130822910455, 7140071740062, 130822910455, 169040786, 32495, 1

Offset

0,5

Comments

Row sums are: {1, 2, 102, 912, 101682, 2278164, 480021360, 20944097328, ...}.

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)*(3*n^2+9*n+5)/30 = A000538(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)*(3*n^2+9*n+5)/30)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.

Example

Triangle begins as:
  1;
  1,     1;
  1,   100,        1;
  1,   455,      455,           1;
  1,  1435,    98810,        1435,           1;
  1,  3711,  1135370,     1135370,        3711,        1;
  1,  8388,  7849141,   464306300,     7849141,     8388,     1;
  1, 17161, 40410421, 10431621081, 10431621081, 40410421, 17161, 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)*(3*n^2+9*n+5)/30)*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)*(3*n^2+9*n+5)/30)*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)*(3*n^2+9*n+5)/30
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)*(3*n^2+9*n+5)/30 >;
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, A154228, A154229, A154231, A154233.

Cf. A000538.

Sequence in context: A284184 A283066 A267440 * A173183 A172179 A147804

Adjacent sequences: A154227 A154228 A154229 * A154231 A154232 A154233

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Jan 05 2009

Extensions

Edited by G. C. Greubel, Mar 02 2021

Status

approved