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A271567 Convolution of nonzero triangular numbers (A000217) and nonzero tetradecagonal numbers (A051866). 0
1, 17, 87, 287, 742, 1638, 3234, 5874, 9999, 16159, 25025, 37401, 54236, 76636, 105876, 143412, 190893, 250173, 323323, 412643, 520674, 650210, 804310, 986310, 1199835, 1448811, 1737477, 2070397, 2452472, 2888952, 3385448, 3947944, 4582809, 5296809, 6097119 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

More generally, the ordinary generating function for the convolution of triangular numbers and k-gonal numbers is (1 + (k - 3)*x)/(1 - x)^6.

LINKS

Table of n, a(n) for n=0..34.

OEIS Wiki, Figurate numbers

Eric Weisstein's World of Mathematics, Triangular Number

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

FORMULA

O.g.f.: (1 + 11*x)/(1 - x)^6.

E.g.f.: (120 + 1920*x + 3240*x^2 + 1520*x^3 + 245*x^4 + 12*x^5)*exp(x)/120.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

a(n) = (n + 1)*(n + 2)*(n + 3)*(n + 4)*(12*n + 5)/120.

Sum_{n>=0} 1/a(n) = 20*((15552*(6*log(2) + 3*log(3) + 2*sqrt(3)*log(2 - sqrt(3)) + (2 - sqrt(3))*Pi) - 29449)/531867) = 1.07654258697...

MATHEMATICA

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 17, 87, 287, 742, 1638}, 40]

Table[(n + 1) (n + 2) (n + 3) (n + 4) (12 n + 5)/120, {n, 0, 40}]

PROG

(MAGMA) /* From definition: */ P:=func<n, k | (n^2*(k-2)-n*(k-4))/2>; /*, where P(n, k) is the n-th k-gonal number, */ [&+[P(n+1-i, 3)*P(i, 14): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Apr 18 2016

(MAGMA) [(n+1)*(n+2)*(n+3)*(n+4)*(12*n+5)/120: n in [0..40]]; // Bruno Berselli, Apr 18 2016

CROSSREFS

Cf. A000217, A051866.

Cf. similar sequences of the convolution of triangular numbers and k-gonal numbers: A005585 (k=4), A051836 (k=5), A034263 (k=6), A027800 (k=7), A051843 (k=8), A051877 (k=9), A051878 (k=10), A051879 (k=11), A051880 (k=12), A056118 (k=13), this sequence (k=14).

Sequence in context: A118863 A118533 A061679 * A231704 A033654 A276318

Adjacent sequences:  A271564 A271565 A271566 * A271568 A271569 A271570

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Apr 12 2016

EXTENSIONS

Edited by Bruno Berselli, Apr 18 2016

STATUS

approved

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Last modified February 20 04:39 EST 2018. Contains 299358 sequences. (Running on oeis4.)