A302355
a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of the g.f. of triangular numbers (A000217).
Original entry on oeis.org
1, 6, 72, 1390, 37515, 1307691, 56000728, 2847503268, 167737660533, 11236731677941, 843757483026150, 70200772129462767, 6410711453857626149, 637516967943664853331, 68581800216461580653064, 7935677122691714769565104, 982824624566131043920711329, 129722104862557293606783635718
Offset: 1
The initial coefficients of successive iterations of g.f. A(x) = x/(1 - x)^3 are as follows:
n = 1: 0, (1), 3, 6, 10, 15, ... g.f. A(x)
n = 2: 0, 1, (6), 30, 137, 588, ... g.f. A(A(x))
n = 3: 0, 1, 9, (72), 543, 3933, ... g.f. A(A(A(x)))
n = 4: 0, 1, 12, 132, (1390), 14208, ... g.f. A(A(A(A(x))))
n = 5: 0, 1, 15, 210, 2840, (37515), ... g.f. A(A(A(A(A(x)))))
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Table[SeriesCoefficient[Nest[Function[x, x/(1 - x)^3], x, n], {x, 0, n}], {n, 18}]
A276644
Self-composition of the repunits; g.f.: A(x) = G(G(x)), where G(x) = g.f. of A002275.
Original entry on oeis.org
0, 1, 22, 464, 9658, 199666, 4112922, 84558014, 1736623658, 35646098566, 731452470122, 15006822709814, 307859627711658, 6315326642698966, 129547066718721322, 2657377349777550614, 54509922224486463658, 1118139793621467673366, 22935894163202834676522, 470473020119757115115414
Offset: 0
Cf.
A030267 (self-composition of the natural numbers),
A030279 (self-composition of the squares),
A030280 (self-composition of the triangular numbers).
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I:=[0,1,22,464]; [n le 4 select I[n] else 33*Self(n-1)-272*Self(n-2)+330*Self(n-3)-100*Self(n-4): n in [1..20]]; // Vincenzo Librandi, Sep 09 2016
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LinearRecurrence[{33, -272, 330, -100}, {0, 1, 22, 464}, 20]
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concat(0, Vec(x*(1-x)*(1-10*x)/((1-21*x+10*x^2)*(1-12*x+10*x^2)) + O(x^99))) \\ Altug Alkan, Sep 08 2016
A279283
Self-composition of the tetrahedral (or triangular pyramidal) numbers; g.f.: A(x) = G(G(x)), where G(x) = g.f. of A000292.
Original entry on oeis.org
0, 1, 8, 52, 304, 1650, 8492, 42000, 201356, 941367, 4310480, 19395042, 85972228, 376185250, 1627518840, 6971209090, 29595604656, 124648174343, 521225809112, 2165408553994, 8942942384500, 36733935375275, 150138939637144, 610840125062072, 2474686297520984, 9986301300789540
Offset: 0
- N. J. A. Sloane, Transforms
- Eric Weisstein's World of Mathematics, Tetrahedral Number
- Index to sequences related to pyramidal numbers
- Index entries for linear recurrences with constant coefficients, signature (20,-174,876,-2885,6708,-11612,15476,-16206,13468,-8894,4632,-1868,564,-120,16,-1).
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CoefficientList[Series[x (1 - x)^12/(1 - 5 x + 6 x^2 - 4 x^3 + x^4)^4, {x, 0, 25}], x]
LinearRecurrence[{20,-174,876,-2885,6708,-11612,15476,-16206,13468,-8894,4632,-1868,564,-120,16,-1},{0,1,8,52,304,1650,8492,42000,201356,941367,4310480,19395042,85972228,376185250,1627518840,6971209090},40] (* Harvey P. Dale, Jul 26 2018 *)
A279284
Self-composition of the pentagonal numbers; g.f.: A(x) = G(G(x)), where G(x) = g.f. of A000326.
Original entry on oeis.org
0, 1, 10, 74, 469, 2662, 14115, 71360, 348143, 1652200, 7669883, 34969286, 157060011, 696514465, 3055404733, 13277356490, 57222978070, 244831062184, 1040760406476, 4398642943496, 18493603597214, 77388169532299, 322451025667910, 1338291853544522, 5534486308363461, 22812231761335189, 93741611639348947, 384122032722040412
Offset: 0
- N. J. A. Sloane, Transforms
- Eric Weisstein's World of Mathematics, Pentagonal Number
- Index to sequences related to polygonal numbers
- Index entries for linear recurrences with constant coefficients, signature (12,-51,91,-75,66,-28,15,-3,1).
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CoefficientList[Series[x (1 - x)^3 (1 + 2 x) (1 - x + 7 x^2 - x^3)/(1 - 4 x + x^2 - x^3)^3, {x, 0, 25}], x]
LinearRecurrence[{12, -51, 91, -75, 66, -28, 15, -3, 1}, {0, 1, 10, 74, 469, 2662, 14115, 71360, 348143}, 26]
Showing 1-4 of 4 results.