A276042 Exponential convolution of central polygonal numbers (A000124) with themselves.
1, 4, 16, 62, 230, 812, 2728, 8752, 26944, 80000, 230144, 644096, 1759744, 4707328, 12359680, 31920128, 81231872, 204013568, 506331136, 1243217920, 3022913536, 7285243904, 17415274496, 41321234432, 97370767360, 227993976832, 530713673728
Offset: 0
Links
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
- Index entries for sequences related to centered polygonal numbers
- Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32)
Programs
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Mathematica
LinearRecurrence[{10, -40, 80, -80, 32}, {1, 4, 16, 62, 230}, 27] Table[2^(n - 6) (n^4 + 2 n^3 + 19 n^2 + 42 n + 64), {n, 0, 26}]
Formula
O.g.f.: (1 - 6*x + 16*x^2 - 18*x^3 + 10*x^4)/(1 - 2*x)^5.
E.g.f.: (2 + 2*x + x^2)^2*exp(2*x)/4.
a(n) = 10*a(n-1) - 40*a(n-2) + 80*a(n-3) - 80*a(n-4) + 32*a(n-5).
a(n) = 2^(n - 6)*(n^4 + 2*n^3 + 19*n^2 + 42*n + 64).