A055268 a(n) = (11*n + 4)*C(n+3, 3)/4.

1, 15, 65, 185, 420, 826, 1470, 2430, 3795, 5665, 8151, 11375, 15470, 20580, 26860, 34476, 43605, 54435, 67165, 82005, 99176, 118910, 141450, 167050, 195975, 228501, 264915, 305515, 350610, 400520, 455576, 516120, 582505, 655095, 734265

Offset: 0

Author

Barry E. Williams, May 10 2000

Keywords

• nonn
• easy

Comments

a(n) is the number of compositions of n when there are 9 types of each natural number. - Milan Janjic, Aug 13 2010

Convolution of A000027 with A051865 (excluding 0). - Bruno Berselli, Dec 07 2012

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
• E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
• Index to sequences related to pyramidal numbers
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Partial sums of A050441.

Cf. A000292, A051865.

Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

Programs

GAP

List([0..30], n-> (11*n+4)*Binomial(n+3,3)/4 ); # G. C. Greubel, Jan 17 2020

Magma

/* A000027 convolved with A051865 (excluding 0): */ A051865:=func; [&+[(n-i+1)*A051865(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012

Maple

seq( (11*n+4)*binomial(n+3,3)/4, n=0..30); # G. C. Greubel, Jan 17 2020

Mathematica

Table[11*Binomial[n+4,4] -10*Binomial[n+3,3], {n,0,30}] (* G. C. Greubel, Jan 17 2020 *)

PARI

a(n) = (11*n+4)*binomial(n+3, 3)/4; \\ Michel Marcus, Sep 07 2017

Python

A055268_list, m = [], [11, 1, 1, 1, 1]
for _ in range(10**2):
    A055268_list.append(m[-1])
    for i in range(4):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

Sage

[(11*n+4)*binomial(n+3,3)/4 for n in (0..30)] # G. C. Greubel, Jan 17 2020

Formula

G.f.: (1 + 10*x)/(1-x)^5. - R. J. Mathar, Oct 26 2011

From G. C. Greubel, Jan 17 2020:(Start)

a(n) = 11*binomial(n+4,4) - 10*binomial(n+3,3).

E.g.f.: (24 + 336*x + 432*x^2 + 136*x^3 + 11*x^4)*exp(x)/24. (End)