A135853 -id:A135853 - cp's OEIS frontend

A135852 A007318 * A103516 as a lower triangular matrix.

1, 3, 2, 8, 4, 3, 20, 6, 9, 4, 48, 8, 18, 16, 5, 112, 10, 30, 40, 25, 6, 256, 12, 45, 80, 75, 36, 7, 576, 14, 63, 140, 175, 126, 49, 8, 1280, 16, 84, 224, 350, 336, 196, 64, 9, 2816, 18, 108, 336, 630, 756, 588, 288, 81, 10

Offset: 0

Gary W. Adamson, Dec 01 2007

Binomial transform of triangle A103516.

			First few rows of the triangle are:
    1;
    3,  2;
    8,  4,  3;
   20,  6,  9,  4;
   48,  8, 18, 16,  5;
  112, 10, 30, 40, 25,  6;
  256, 12, 45, 80, 75, 36,  7;
  ...

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

Cf. A007318, A103516.
Cf. A001792 (1st column), A099035 (row sums).
Cf. A135853 (= A103516 * A007318).

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

def A135852(n,k):
    if (n==0): return 1
    elif (k==0): return (n+2)*2^(n-1)
    else: return (k+1)*binomial(n, k)
flatten([[A135852(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 07 2022

T(n, k) = (A007318 * A103516)(n, k).
T(n, 0) = A001792(n).
Sum_{k=0..n} T(n, k) = A099035(n+1).
T(n, k) = (k+1)*binomial(n, k), with T(n, 0) = (n+2)*2^(n-1), T(n, n) = n+1. - G. C. Greubel, Dec 07 2016

Offset changed to 0 by G. C. Greubel, Feb 07 2022

A135854 a(n) = (n+1)*(2^n+1) for n > 0 with a(0)=1.

Original entry on oeis.org

1, 6, 15, 36, 85, 198, 455, 1032, 2313, 5130, 11275, 24588, 53261, 114702, 245775, 524304, 1114129, 2359314, 4980755, 10485780, 22020117, 46137366, 96469015, 201326616, 419430425, 872415258, 1811939355, 3758096412

Offset: 0

Gary W. Adamson, Dec 01 2007

			a(3) = 15 = sum of row 3 terms of triangle A135853: (6 + 6 + 3).
a(4) = 36 = (1, 3, 3, 1) dot (1, 5, 4, 8) = (1 + 15 + 12 + 8).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4)

Cf. A215149.

Row sums of triangle A135853.

A135854:=n->(n+1)*(2^n+1): 1, seq(A135854(n), n=1..50); # Wesley Ivan Hurt, Dec 07 2016

Join[{1}, LinearRecurrence[{6,-13,12,-4}, {6,15,36,85}, 25]] (* G. C. Greubel, Dec 07 2016 *)

Vec((1-8*x^2+12*x^3-4*x^4)/((1-x)^2*(1-2*x)^2) + O(x^50)) \\ G. C. Greubel, Dec 07 2016

Binomial transform of [1, 5, 4, 8, 8, 12, 12, 16, 16, 20, 20, ...].
G.f.: 1 - x*(-6 + 21*x - 24*x^2 + 8*x^3) / ( (2*x-1)^2*(x-1)^2 ). - R. J. Mathar, Apr 04 2012
G.f.: (1 - 8*x^2 + 12*x^3 - 4*x^4)/((1-x)^2*(1-2*x)^2). - L. Edson Jeffery, Jan 14 2014
a(0) = 1, a(n) = (n+1)*(2^n+1), n>0. - L. Edson Jeffery, Jan 14 2014
E.g.f.: exp(x)*(1 + x + exp(x)*(1 + 2*x)) - 1. - Stefano Spezia, Dec 13 2021

Corrected by R. J. Mathar, Apr 04 2012

cp's OEIS Frontend

A135852 A007318 * A103516 as a lower triangular matrix.

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