A287143 Expansion of x*(1 + 3*x + x^2)/((1 - x)^5*(1 + x)^4).

0, 1, 4, 9, 21, 35, 65, 95, 155, 210, 315, 406, 574, 714, 966, 1170, 1530, 1815, 2310, 2695, 3355, 3861, 4719, 5369, 6461, 7280, 8645, 9660, 11340, 12580, 14620, 16116, 18564, 20349, 23256, 25365, 28785, 31255, 35245, 38115, 42735, 46046, 51359, 55154, 61226, 65550, 72450, 77350, 85150, 90675, 99450, 105651, 115479

Offset: 0

Author

Ilya Gutkovskiy, Aug 15 2017

Keywords

• nonn
• easy

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. A002413, A002418, A002419, A002624, A027656, A085787, A212246, A290055.

Programs

Mathematica

CoefficientList[Series[x (1 + 3 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 52}], x]
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 4, 9, 21, 35, 65, 95, 155}, 53]

Formula

G.f.: x*(1 + 3*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 (A002418): (5*n - 1)*binomial(n + 2,3)/4, n = 0,+1,-3,+2,-4,+3,-5, ...

Convolution of the sequences A027656 and A085787.

a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(5*(2*n^2+10*n+3)-3*(2*n+5)*(-1)^n)/3072. - Luce ETIENNE, Nov 18 2017