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A276659 Accumulation of the upper left triangle used in binomial transform of nonnegative integers. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A259213 A259658 A000175 * A187259 A296593 A125064

Adjacent sequences:  A276656 A276657 A276658 * A276660 A276661 A276662

KEYWORD

nonn,easy

AUTHOR

Jean-Fran├žois Alcover and Francois Alcover, Sep 11 2016

EXTENSIONS

Edited and extended by Bruno Berselli, Sep 13 2016

STATUS

approved

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Last modified June 25 02:23 EDT 2021. Contains 345445 sequences. (Running on oeis4.)