A276659 Accumulation of the upper left triangle used in binomial transform of nonnegative integers.

0, 2, 11, 39, 114, 300, 741, 1757, 4052, 9162, 20415, 44979, 98214, 212888, 458633, 982905, 2097000, 4456278, 9436995, 19922735, 41942810, 88080132, 184549101, 385875669, 805306044, 1677721250, 3489660551, 7247756907, 15032385102, 31138512432, 64424508945

Offset: 0

Author

Jean-François Alcover and Francois Alcover, Sep 11 2016

Keywords

• nonn
• easy

Comments

After 0, is this the second column of A108284? [Bruno Berselli, Sep 13 2016 - this comment may be removed if the property is confirmed.]

Examples

			Starting from the triangle:
   0,  1,  2,  3,  4,  5, ...
   1,  3,  5,  7,  9, ...
   4,  8, 12, 16, ...
  12, 20, 28, ...
  32, 48, ...
  80, ...
  ...
the first terms are:
a(0) = 0;
a(1) = a(0) + 1 + 1 = 2;
a(2) = a(1) + 4 + 3 + 2 = 11;
a(3) = a(2) + 12 + 8 + 5 + 3 = 39, etc.
First column is A001787: n*2^(n-1).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (7,-19,25,-16,4).

Crossrefs

Cf. A001787, A058877, A062111, A152920.

Programs

Magma

[(2^(n+2)-n-3)*n/2: n in [0..40]]; // Vincenzo Librandi, Sep 13 2016

Maple

A276659:=n->n*(2^(n+2) - n - 3)/2: seq(A276659(n), n=0..50); # Wesley Ivan Hurt, Sep 16 2017

Mathematica

t[0, k_] := k; t[n_, k_] := t[n, k] = t[n - 1, k] + t[n - 1, k + 1]; a[n_] := Sum[t[m, k], {m, 0, n}, {k, 0, n - m}]; Table[a[n], {n, 0, 30}]
Table[(2^(n + 2) - n - 3) n / 2, {n, 0, 30}] (* Vincenzo Librandi, Sep 13 2016 *)

PARI

x='x+O('x^99); concat(0, Vec(x*(2-3*x)/((1-x)^3*(1-2*x)^2))) \\ Altug Alkan, Sep 14 2017

Formula

O.g.f.: x*(2 - 3*x)/((1 - x)^3*(1 - 2*x)^2).

E.g.f.: x*exp(x)*(8*exp(x) - x - 4)/2.

a(n) = n*(2^(n+2) - n - 3)/2.

a(n) = 7*a(n-1) - 19*a(n-2) + 25*a(n-3) - 16*a(n-4) + 4*a(n-5) for n > 4.

a(n) = a(n-1) + A058877(n+1). - R. J. Mathar, Sep 14 2016

a(n) = Sum_{k=2..n+3} Sum_{i=2..n+3} k * C(n-i+3,k). - Wesley Ivan Hurt, Sep 20 2017

Extensions

Edited and extended by Bruno Berselli, Sep 13 2016