A002419 4-dimensional figurate numbers: a(n) = (6*n-2)*binomial(n+2,3)/4.

1, 10, 40, 110, 245, 476, 840, 1380, 2145, 3190, 4576, 6370, 8645, 11480, 14960, 19176, 24225, 30210, 37240, 45430, 54901, 65780, 78200, 92300, 108225, 126126, 146160, 168490, 193285, 220720, 250976, 284240, 320705, 360570, 404040, 451326, 502645, 558220

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the n-th antidiagonal sum of the convolution array A213761. - Clark Kimberling, Jul 04 2012

Convolution of A000027 with A000567 (excluding 0). - Bruno Berselli, Dec 07 2012

a(n) = the sum of all the ways of adding the k-tuples of A016777(0) to A016777(n-1). For n=4, the terms are 1,4,7,10 giving (1)+(4)+(7)+(10)=22; (1+4)+(4+7)+(7+10)=33; (1+4+7)+(4+7+10)=33; (1+4+7+10)=22; adding 22+33+33+22=110. - J. M. Bergot, Jun 26 2017

Also the number of chordless cycles in the (n+2)-crown graph. - Eric W. Weisstein, Jan 02 2018

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Eric Weisstein's World of Mathematics, Chordless Cycle.
• Eric Weisstein's World of Mathematics, Crown Graph.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
• Index to sequences related to pyramidal numbers.

Crossrefs

Cf. A093563 ((6, 1) Pascal, column m=4).

Cf. A000027, A000567, A002414 (first differences), A016777, A080852, A213761.

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

Programs

GAP

List([1..40], n-> n*(n+1)*(n+2)*(3*n-1)/12); # G. C. Greubel, Jul 03 2019

Magma

/* A000027 convolved with A000567 (excluding 0): */ A000567:=func; [&+[(n-i+1)*A000567(i): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Dec 07 2012

Mathematica

CoefficientList[Series[(1+5*x)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Jun 20 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 10, 40, 110, 245}, 40] (* Harvey P. Dale, Nov 30 2014 *)
Table[n(n+1)(n+2)(3n-1)/12, {n, 40}] (* Eric W. Weisstein, Jan 02 2018 *)
Table[Sum[2 x + 3 x^2 - 2 y, {x, 0, g}, {y, x, g}], {g, 1, 20}] (* Horst H. Manninger, Jun 20 2025 *)

PARI

a(n)=(3*n-1)*binomial(n+2,3)/2 \\ Charles R Greathouse IV, Sep 24 2015

Python

A002419_list, m = [], [6, 1, 1, 1, 1]
for _ in range(10**2):
    A002419_list.append(m[-1])
    for i in range(4):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

Sage

[n*(n+1)*(n+2)*(3*n-1)/12 for n in (1..40)] # G. C. Greubel, Jul 03 2019

Formula

a(n) = (3*n-1)*binomial(n+2, 3)/2.

G.f.: x*(1+5*x)/(1-x)^5. - Simon Plouffe in his 1992 dissertation.

Sum_{n>=1} 1/a(n) = (-24+81*log(3) -9*Pi*sqrt(3))/14 = 1.143929... - R. J. Mathar, Mar 29 2011

a(n) = (3*n^4 + 8*n^3 + 3*n^2 - 2*n)/12. - Chai Wah Wu, Jan 24 2016

a(n) = A080852(6,n-1). - R. J. Mathar, Jul 28 2016

E.g.f.: x*(12 + 48*x + 26*x^2 + 3*x^3)*exp(x)/12. - G. C. Greubel, Jul 03 2019

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(3*sqrt(3)*Pi - 32*log(2) + 8)/7. - Amiram Eldar, Feb 11 2022