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1, 12, 50, 140, 315, 616, 1092, 1800, 2805, 4180, 6006, 8372, 11375, 15120, 19720, 25296, 31977, 39900, 49210, 60060, 72611, 87032, 103500, 122200, 143325, 167076, 193662, 223300, 256215, 292640, 332816, 376992, 425425, 478380, 536130
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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REFERENCES
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Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
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LINKS
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FORMULA
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a(n) = binomial(n+3,3)*(2*n+1) = (n+1)*(n+2)*(n+3)*(2*n+1)/6.
G.f.: (1+7*x)/(1-x)^5.
E.g.f.: (6 + 66*x + 81*x^2 + 25*x^3 + 2*x^4)*exp(x)/6. - G. C. Greubel, Aug 30 2019
Sum_{n>=0} 1/a(n) = (32*log(2) - 11)/10.
Sum_{n>=0} (-1)^n/a(n) = (8*Pi - 56*log(2) + 23)/10. (End)
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MAPLE
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seq((2*n+1)*binomial(n+3, 3), n=0..40); # G. C. Greubel, Aug 30 2019
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MATHEMATICA
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PROG
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(Magma) [(2*n+1)*Binomial(n+3, 3): n in [0..40]]; // G. C. Greubel, Aug 30 2019
(PARI) vector(40, n, (2*n-1)*binomial(n+2, 3)) \\ G. C. Greubel, Aug 30 2019
(Sage) [(2*n+1)*binomial(n+3, 3) for n in (0..40)] # G. C. Greubel, Aug 30 2019
(GAP) List([0..40], n-> (2*n+1)*Binomial(n+3, 3)) # G. C. Greubel, Aug 30 2019
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CROSSREFS
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Cf. A093565 ((8, 1) Pascal, column m=4).
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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