A002418 4-dimensional figurate numbers: a(n) = (5*n-1)*binomial(n+2,3)/4. (Formerly M4617 N1970)

0, 1, 9, 35, 95, 210, 406, 714, 1170, 1815, 2695, 3861, 5369, 7280, 9660, 12580, 16116, 20349, 25365, 31255, 38115, 46046, 55154, 65550, 77350, 90675, 105651, 122409, 141085, 161820, 184760, 210056, 237864, 268345, 301665, 337995

Offset

0,3

Comments

Partial sums of A002413.

Principal diagonal of the convolution array A213550, for n>0. - Clark Kimberling, Jun 17 2012

Convolution of A000027 with A000566. - Bruno Berselli, Dec 06 2012

Coefficients in the hypergeometric series identity 1 - 9*(x - 1)/(4*x + 1) + 35*(x - 1)*(x - 2)/((4*x + 1)*(4*x + 2)) - 95*(x - 1)*(x - 2)*(x - 3)/((4*x + 1)*(4*x + 2)*(4*x + 3)) + ... = 0, valid for Re(x) > 1. Cf. A000326 and A002412. Column 4 of A103450. - Peter Bala, Mar 14 2019

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 = 0..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.

Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5, 1).

Index to sequences related to pyramidal numbers.

Formula

G.f.: x*(1+4*x)/(1-x)^5. - Simon Plouffe in his 1992 dissertation.

Starting (1, 9, 35, 95, ...), = A128064 * A000332, (A000332 starting 1, 5, 15, 35, 70, ...), such that a(n) = n*C(n+3,4) - (n-1)*C(n+2,4). E.g., a(5) = 210 = 5*C(8,4) - 4*C(7,4) = 5*70 - 4*35. - Gary W. Adamson, Dec 28 2007

Unit digit, A010879(a(n)), is one of {0,1,9,5,6,4} [Eric Desbiaux] because a(n) mod 5 = 0,1,4,0,0, periodic with period 5. [Proof: A002413(n) mod 5 = 1,3,1,0,0 with period 5 and a(n) are the partial sums of A002413.] - R. J. Mathar, Mar 19 2008

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Oct 16 2012

a(n) = A080852(5,n-1). - R. J. Mathar, Jul 28 2016

a(n) = Sum_{i=0..n} (n-i) * Sum_{j=i..n} j. - J. M. Bergot, May 30 2017

E.g.f.: x*(24 + 84*x + 44*x^2 + 5*x^3)*exp(x)/4!. - G. C. Greubel, Jul 03 2019

Sum_{n>=1} 1/a(n) = (50*sqrt(5)*log(phi) + 125*log(5) - 50*sqrt(1+2/sqrt(5))*Pi - 26)/11, where phi is the golden ratio (A001622). - Amiram Eldar, Feb 11 2022

Mathematica

Table[(5n-1) Binomial[n+2, 3]/4, {n, 0, 40}] (* or *) LinearRecurrence[ {5, -10, 10, -5, 1}, {0, 1, 9, 35, 95}, 40] (* Harvey P. Dale, Oct 16 2012 *)
CoefficientList[Series[x*(1 + 4*x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 17 2012 *)

Prog

(Magma) [(5*n - 1)*Binomial(n + 2, 3)/4: n in [0..40]]; // Vincenzo Librandi, Oct 17 2012
(Magma) /* A000027 convolved with A000566: */ A000566:=func<n | n*(5*n-3)/2>; [&+[(n-i+1)*A000566(i): i in [0..n]]: n in [0..35]]; // Bruno Berselli, Dec 06 2012
(PARI) a(n)=(5*n-1)*binomial(n+2, 3)/4 \\ Charles R Greathouse IV, Sep 24 2015
(GAP) List([0..40], n->(5*n-1)*Binomial(n+2, 3)/4); # Muniru A Asiru, Mar 18 2019
(Sage) [(5*n-1)*binomial(n+2, 3)/4 for n in (0..40)] # G. C. Greubel, Jul 03 2019

Crossrefs

Cf. A093562 ((5, 1) Pascal, column m=4).

Cf. A000332, A000566, A001622, A080852, A128064.

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

Sequence in context: A005898 A034957 A180082 * A118414 A279218 A322239

Adjacent sequences: A002415 A002416 A002417 * A002419 A002420 A002421

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved