A135853 A103516 * A007318 as an infinite lower triangular matrix.

1, 4, 2, 6, 6, 3, 8, 12, 12, 4, 10, 20, 30, 20, 5, 12, 30, 60, 60, 30, 6, 14, 42, 105, 140, 105, 42, 7, 16, 56, 168, 280, 280, 168, 56, 8, 18, 72, 252, 504, 630, 504, 252, 72, 9, 20, 90, 360, 840, 1260, 1260, 840, 360, 90, 10

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows

Formula

T(n, k) = (A103516 * A007318)(n, k).

Sum_{k=0..n} T(n, k) = A135854(n).

T(n, k) = (k+1)*binomial(n+1, k+1), with T(n, n) = n+1, T(n, 0) = 2*(n+1). - G. C. Greubel, Dec 07 2016

T(n, 0) = A103517(n). - G. C. Greubel, Feb 06 2022

Example

First few rows of the triangle are:
   1;
   4,   2;
   6,   6,  3;
   8,  12,  12,   4;
  10,  20,  30,  20,   5;
  12,  30,  60,  60,  30,   6;
  14,  42, 105, 140, 105,  42,   7;
  ...

Mathematica

T[n_, k_]:= If[k==n, n+1, If[k==0, 2*(n+1), (k+1)*Binomial[n+1, k+1]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//flatten (* G. C. Greubel, Dec 07 2016 *)

Prog

(Sage)
def A135853(n, k):
    if (n==0): return 1
    elif (k==0): return 2*(n+1)
    else: return (k+1)*binomial(n+1, k+1)
flatten([[A135853(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 06 2022

Crossrefs

Cf. A007318, A103516.

Cf. A103517 (1st column), A135854 (row sums).

Cf. A135852 (= A007318 * A103516).

Sequence in context: A344123 A188941 A200347 * A338915 A173197 A256568

Adjacent sequences: A135850 A135851 A135852 * A135854 A135855 A135856

Keyword

nonn,tabl

Author

Gary W. Adamson, Dec 01 2007

Status

approved