A157171 Triangle, read by rows, T(n,k,m) = (m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1), with m = 3.

1, 1, 1, 1, 11, 1, 1, 21, 21, 1, 1, 31, 66, 31, 1, 1, 41, 136, 136, 41, 1, 1, 51, 231, 362, 231, 51, 1, 1, 61, 351, 755, 755, 351, 61, 1, 1, 71, 496, 1361, 1870, 1361, 496, 71, 1, 1, 81, 666, 2226, 3906, 3906, 2226, 666, 81, 1, 1, 91, 861, 3396, 7266, 9282, 7266, 3396, 861, 91, 1

Offset

0,5

Comments

Row sums are: {1, 2, 13, 44, 130, 356, 928, 2336, 5728, 13760, 32512, ...}.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n,k,m) = (m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1), with m = 3.

Example

Triangle begins as:
  1;
  1,  1;
  1, 11,   1;
  1, 21,  21,    1;
  1, 31,  66,   31,    1;
  1, 41, 136,  136,   41,    1;
  1, 51, 231,  362,  231,   51,    1;
  1, 61, 351,  755,  755,  351,   61,    1;
  1, 71, 496, 1361, 1870, 1361,  496,   71,   1;
  1, 81, 666, 2226, 3906, 3906, 2226,  666,  81,  1;
  1, 91, 861, 3396, 7266, 9282, 7266, 3396, 861, 91, 1;

Maple

T:= proc(n, k, m) option remember;
      if k=0 and n=0 then 1
    else (m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1)
      fi; end:
seq(seq(T(n, k, 3), k=0..n), n=0..10); # G. C. Greubel, Nov 29 2019

Mathematica

T[n_, k_, m_]:= If[n==0 && k==0, 1, (m*(n-k)+1)*Binomial[n-1, k-1] + (m*k+1)*Binomial[n-1, k] + m*k*(n-k)*Binomial[n-2, k-1]]; Table[T[n, k, 3], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Nov 29 2019 *)

Prog

(PARI) T(n, k, m) = (m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1); \\ G. C. Greubel, Nov 29 2019
(Magma) m:=3; [(m*(n-k)+1)*Binomial(n-1, k-1) + (m*k+1)* Binomial(n-1, k) + m*k*(n-k)*Binomial(n-2, k-1): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 29 2019
(Sage) m=3; [[(m*(n-k)+1)*binomial(n-1, k-1) + (m*k+1)* binomial(n-1, k) + m*k*(n-k)*binomial(n-2, k-1) for k in (0..n)] for n in [0..10]] # G. C. Greubel, Nov 29 2019
(GAP) m:=3;; Flat(List([0..10], n-> List([0..n], k-> (m*(n-k)+1)*Binomial(n-1, k-1) + (m*k+1)* Binomial(n-1, k) + m*k*(n-k)*Binomial(n-2, k-1) ))); # G. C. Greubel, Nov 29 2019

Crossrefs

Cf. A157169 (m=1), A157170 (m=2), this sequence (m=3).

Sequence in context: A010192 A214326 A105769 * A143685 A168647 A202767

Adjacent sequences: A157168 A157169 A157170 * A157172 A157173 A157174

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 24 2009

Extensions

Edited by G. C. Greubel, Nov 29 2019

Status

approved