A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

0, 1, 16, 70, 200, 455, 896, 1596, 2640, 4125, 6160, 8866, 12376, 16835, 22400, 29240, 37536, 47481, 59280, 73150, 89320, 108031, 129536, 154100, 182000, 213525, 248976, 288666, 332920, 382075, 436480, 496496, 562496, 634865, 714000, 800310, 894216, 996151

Offset

0,3

Comments

Partial sums of A172073.

Apart from 0, all terms are in A135021: a(n) = A135021(A034856(n+1)) with n>0.

Links

Bruno Berselli, Table of n, a(n) for n = 0..1000

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+11*x)/(1-x)^5.

a(n) = n*(n+1)*(n+2)*(3*n-2)/6.

From Amiram Eldar, Feb 15 2022: (Start)

Sum_{n>=1} 1/a(n) = 3*(3*sqrt(3)*Pi + 27*log(3) - 17)/80.

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(6*sqrt(3)*Pi - 64*log(2) + 37)/80. (End)

Mathematica

A051866[k_] := k (6 k - 5); Table[Sum[(n - k + 1) A051866[k], {k, 0, n}], {n, 0, 37}]
CoefficientList[Series[x (1 + 11 x) / (1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Prog

(Magma) A051866:=func<n | n*(6*n-5)>; [&+[(n-k+1)*A051866(k): k in [0..n]]: n in [0..37]];
(Magma) I:=[0, 1, 16, 70, 200]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

Crossrefs

Cf. A135021, A172073.

Cf. convolution of the natural numbers (A000027) with the k-gonal numbers (* means "except 0"):

k= 2 (A000027 ): A000292;

k= 3 (A000217 ): A000332 (after the third term);

k= 4 (A000290 ): A002415 (after the first term);

k= 5 (A000326 ): A001296;

k= 6 (A000384*): A002417;

k= 7 (A000566 ): A002418;

k= 8 (A000567*): A002419;

k= 9 (A001106*): A051740;

k=10 (A001107*): A051797;

k=11 (A051682*): A051798;

k=12 (A051624*): A051799;

k=13 (A051865*): A055268.

Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.

Sequence in context: A200839 A036660 A063493 * A027997 A284844 A258724

Adjacent sequences: A220209 A220210 A220211 * A220213 A220214 A220215

Keyword

nonn,easy

Author

Bruno Berselli, Dec 08 2012

Status

approved