A127896
Expansion of 1/(1 + 2*x + 3*x^2 + x^3).
Original entry on oeis.org
1, -2, 1, 3, -7, 4, 10, -25, 16, 33, -89, 63, 108, -316, 245, 350, -1119, 943, 1121, -3952, 3598, 3539, -13920, 13625, 10971, -48897, 51256, 33208, -171287, 191694, 97265, -598325, 713161, 271388, -2083934
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
- Index entries for linear recurrences with constant coefficients, signature (-2,-3,-1).
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I:=[1, -2, 1]; [n le 3 select I[n] else -2*Self(n-1) -3*Self(n-2) -Self(n-3): n in [1..50]]; // G. C. Greubel, Apr 29 2018
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CoefficientList[Series[1/(1+2x+3x^2+x^3),{x,0,40}],x] (* Harvey P. Dale, Apr 19 2011 *)
LinearRecurrence[{-2, -3, -1}, {1, -2, 1}, 30] (* G. C. Greubel, Apr 29 2018 *)
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x='x+O('x^50); Vec(1/(1+2*x+3*x^2+x^3)) \\ G. C. Greubel, Apr 29 2018
A127898
Inverse of Riordan array (1/(1+x)^3, x/(1+x)^3).
Original entry on oeis.org
1, 3, 1, 12, 6, 1, 55, 33, 9, 1, 273, 182, 63, 12, 1, 1428, 1020, 408, 102, 15, 1, 7752, 5814, 2565, 760, 150, 18, 1, 43263, 33649, 15939, 5313, 1265, 207, 21, 1, 246675, 197340, 98670, 35880, 9750, 1950, 273, 24, 1
Offset: 0
Triangle begins:
1,
3, 1,
12, 6, 1,
55, 33, 9, 1,
273, 182, 63, 12, 1,
1428, 1020, 408, 102, 15, 1,
7752, 5814, 2565, 760, 150, 18, 1,
43263, 33649, 15939, 5313, 1265, 207, 21, 1,
246675, 197340, 98670, 35880, 9750, 1950, 273, 24, 1,
1430715, 1170585, 610740, 237510, 71253, 16443, 2842, 348, 27, 1,
8414640, 7012200, 3786588, 1553472, 503440, 129456, 26040, 3968, 432, 30, 1
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Flat(List([0..10],n->List([0..n],k->(k+1)/(n+1)*Binomial(3*n+3,n-k)))); # Muniru A Asiru, Apr 30 2018
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/* As triangle: */ [[(k+1)/(n+1)*Binomial(3*n+3,n-k): k in [0..n]]: n in [0..8]]; // Bruno Berselli, Jan 17 2013
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# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, n -> binomial(3*n, n)/(2*n+1)); # Peter Luschny, Oct 09 2022
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Table[If[k == 0, Binomial[3*n, n-k]/(2*n+1), ((k+1)/n)*Binomial[3*n, n-k -1]], {n,1,10}, {k,0,n-1}]//Flatten (* G. C. Greubel, Apr 29 2018 *)
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for(n=1,10, for(k=0,n-1, print1(if(k==0, binomial(3*n, n-k)/( 2*n +1), ((k+1)/n)*binomial(3*n, n-k-1)), ", "))) \\ G. C. Greubel, Apr 29 2018
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