A290055 Expansion of x*(1 + 4*x + x^2)/((1 - x)^5*(1 + x)^4).

0, 1, 5, 10, 26, 40, 80, 110, 190, 245, 385, 476, 700, 840, 1176, 1380, 1860, 2145, 2805, 3190, 4070, 4576, 5720, 6370, 7826, 8645, 10465, 11480, 13720, 14960, 17680, 19176, 22440, 24225, 28101, 30210, 34770, 37240, 42560, 45430, 51590, 54901, 61985, 65780, 73876, 78200, 87400, 92300, 102700, 108225, 119925, 126126

Offset: 0

Author

Ilya Gutkovskiy, Aug 15 2017

Keywords

• nonn
• easy

Comments

More generally, the generalized 4-dimensional figurate numbers are convolution of the sequence {1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...} with generalized polygonal numbers (A195152).

Links

• Index to sequences related to pyramidal numbers
• Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Crossrefs

Cf. A001082, A002414, A002419, A002624, A027656, A195152, A287143.

Programs

Mathematica

CoefficientList[Series[x (1 + 4 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 51}], x]
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 5, 10, 26, 40, 80, 110, 190}, 52]

PARI

x='x+O('x^99); concat(0, Vec(x*(1+4*x+x^2)/((1-x)^5*(1 + x)^4))) \\ Altug Alkan, Aug 15 2017

Formula

G.f.: x*(1 + 4*x + x^2)/((1 - x)^5*(1 + x)^4).

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).

Generalized 4-dimensional figurate numbers (A002419): (3*n - 1)*binomial(n + 2,3)/2, n = 0,+1,-3,+2,-4,+3,-5, ...

Convolution of the sequences A027656 and A001082 (with offset 0).

a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(6*n^2+30*n+5-(2*n+5)*(-1)^n)/1536. - Luce ETIENNE, Nov 18 2017