A329523 a(n) = n * (binomial(n + 1, 3) + 1).

0, 1, 4, 15, 44, 105, 216, 399, 680, 1089, 1660, 2431, 3444, 4745, 6384, 8415, 10896, 13889, 17460, 21679, 26620, 32361, 38984, 46575, 55224, 65025, 76076, 88479, 102340, 117769, 134880, 153791, 174624, 197505, 222564, 249935, 279756, 312169, 347320, 385359, 426440

Offset

0,3

Comments

The n-th centered n-gonal pyramidal number.

References

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), 142.

Links

Table of n, a(n) for n=0..40.

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

Formula

G.f.: x * (1 - x + 5*x^2 - x^3) / (1 - x)^5.

E.g.f.: exp(x) * x * (1 + x + x^2 + x^3 / 6).

a(n) = n * (n + 2) * (n^2 - 2*n + 3) / 6.

a(n) = n * (A000292(n-1) + 1).

a(n) = n + 2 * Sum_{k=1..n} A000330(k-1).

a(n) + a(-n) = 4 * A002415(n).

Example

Square array begins:
  (0), 1,  2,   3,   4,    5,  ... A001477
   0, (1), 3,   7,  14,   25,  ... A004006
   0,  1, (4), 11,  24,   45,  ... A006527
   0,  1,  5, (15), 34,   65,  ... A006003 (partial sums of A005448)
   0,  1,  6,  19, (44),  85,  ... A005900 (partial sums of A001844)
   0,  1,  7,  23,  54, (105), ... A004068 (partial sums of A005891)
...
This sequence is the main diagonal of the array.

Mathematica

Table[n (Binomial[n + 1, 3] + 1), {n, 0, 40}]
nmax = 40; CoefficientList[Series[x (1 - x + 5 x^2 - x^3)/(1 - x)^5, {x, 0, nmax}], x]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 4, 15, 44}, 41]

Prog

(Magma) [ n*(Binomial(n+1, 3)+1):n in [0..40]]; // Marius A. Burtea, Nov 15 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 41); [0] cat Coefficients(R!(x*(1-x+5*x^2-x^3)/(1-x)^5)); // Marius A. Burtea, Nov 15 2019

Crossrefs

Cf. A000292, A000330, A002415, A002417, A006000, A006484, A008911, A050407, A060354, A100119, A188475 (first differences).

Sequence in context: A294259 A240359 A282522 * A331317 A259664 A321880

Adjacent sequences: A329520 A329521 A329522 * A329524 A329525 A329526

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Nov 15 2019

Status

approved