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

0,2

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.]

Links

Table of n, a(n) for n=0..30.

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

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

Example

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).

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 *)

Prog

(Magma) [(2^(n+2)-n-3)*n/2: n in [0..40]]; // 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

Crossrefs

Cf. A001787, A058877, A062111, A152920.

Sequence in context: A259658 A350952 A000175 * A187259 A296593 A125064

Adjacent sequences: A276656 A276657 A276658 * A276660 A276661 A276662

Keyword

nonn,easy

Author

Jean-François Alcover and Francois Alcover, Sep 11 2016

Extensions

Edited and extended by Bruno Berselli, Sep 13 2016

Status

approved