A157174 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, 5, 1, 1, 9, 9, 1, 1, 13, 18, 13, 1, 1, 17, 28, 28, 17, 1, 1, 21, 39, 38, 39, 21, 1, 1, 25, 51, 35, 35, 51, 25, 1, 1, 29, 64, 11, -50, 11, 64, 29, 1, 1, 33, 78, -42, -294, -294, -42, 78, 33, 1, 1, 37, 93, -132, -798, -1218, -798, -132, 93, 37, 1

Offset

0,5

Comments

Row sums are: {1, 2, 7, 20, 46, 92, 160, 224, 160, -448, -28166, ...}.

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,  5,  1;
  1,  9,  9,    1;
  1, 13, 18,   13,    1;
  1, 17, 28,   28,   17,     1;
  1, 21, 39,   38,   39,    21,    1;
  1, 25, 51,   35,   35,    51,   25,    1;
  1, 29, 64,   11,  -50,    11,   64,   29,  1;
  1, 33, 78,  -42, -294,  -294,  -42,   78, 33,  1;
  1, 37, 93, -132, -798, -1218, -798, -132, 93, 37, 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. A157172 (m=2), this sequence (m=3).

Sequence in context: A046583 A046579 A153108 * A183450 A296128 A131061

Adjacent sequences: A157171 A157172 A157173 * A157175 A157176 A157177

Keyword

sign,tabl

Author

Roger L. Bagula and Gary W. Adamson, Feb 24 2009

Extensions

Edited by G. C. Greubel, Nov 29 2019

Status

approved