A286781
Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
Original entry on oeis.org
1, 2, 1, 10, 9, 1, 74, 91, 23, 1, 706, 1063, 416, 46, 1, 8162, 14193, 7344, 1350, 80, 1, 110410, 213953, 134613, 34362, 3550, 127, 1, 1708394, 3602891, 2620379, 842751, 125195, 8085, 189, 1, 29752066, 67168527, 54636792, 20862684, 4009832, 382358, 16576, 268, 1, 576037442, 1375636129, 1223392968, 533394516, 124266346, 15653598, 1023340, 31356, 366, 1
Offset: 0
A(x;t) = 1 + (2 + t)*x + (10 + 9*t + t^2)*x^2 + (74 + 91*t + 23*t^2 + t^3)*x^3 + ...
Triangle starts:
n\k [0] [1] [2] [3] [4] [5] [6] [7] [8]
[0] 1;
[1] 2, 1;
[2] 10, 9, 1;
[3] 74, 91, 23, 1;
[4] 706, 1063, 416, 46, 1;
[5] 8162, 14193, 7344, 1350, 80, 1;
[6] 110410, 213953, 134613, 34362, 3550, 127, 1;
[7] 1708394, 3602891, 2620379, 842751, 125195, 8085, 189, 1;
[8] 29752066, 67168527, 54636792, 20862684, 4009832, 382358, 16576, 268, 1;
[9] ...
For vertex and polarization functions see
A286782 and
A286783. For GWA of the self-energy and polarization functions see
A286784 and
A286785.
-
max = 10; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = (1 + x*y0[x, t] + 2*x^2*D[y0[x, t], x])*(1 - x*y0[x, t]*(1 - t))/(1 - x*y0[x, t])^2 + O[x]^n // Normal; y0[x_, t_] = y1[x, t]];
row[n_] := CoefficientList[Coefficient[y0[x, t], x, n], t];
Table[row[n], {n, 0, max-1}] // Flatten (* Jean-François Alcover, May 19 2017, adapted from PARI *)
-
A286781_ser(N,t='t) = {
my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
while(n++,
y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
if (y1 == y0, break()); y0 = y1;);
y0;
};
concat(apply(p->Vecrev(p), Vec(A286781_ser(10))))
\\ test: y = A286781_ser(50); y*(1-x*y)^2 == (1 + x*y + 2*x^2*deriv(y,'x)) * (1 - x*y*(1-t))
Original entry on oeis.org
1, 46, 1350, 34362, 842751, 20862684, 533394516, 14251251012, 400275122533, 11849888387786, 370032953752618, 12183464636804638, 422529226131247475, 15413172354966378040, 590459747844232859976, 23715530712277152761928, 997006212536288359365609, 43800374567467950834916070
Offset: 3
Original entry on oeis.org
1, 80, 3550, 125195, 4009832, 124266346, 3854188670, 121943460540, 3979525634005, 134806148044750, 4756728585598416, 175132764223880035, 6731931366383154760, 270129431174068396380, 11308903054417009540644, 493544598057755406560688, 22431785171561428611320865
Offset: 4
Original entry on oeis.org
1, 127, 8085, 382358, 15653598, 598415692, 22280470762, 828351390146, 31225895068199, 1204894853397543, 47870402081423775, 1965205314867907972, 83531844407610101388, 3679988322044604002256, 168093873703636990132596, 7960504783782783371445468
Offset: 5
Original entry on oeis.org
1, 189, 16576, 1023340, 52563182, 2445890678, 108248782988, 4693939818672, 203214066766123, 8891156436472127, 396276632116578640, 18084968846773765272, 847923332620865268252, 40927316768759272656876, 2036183873275911481501656, 104482592642320957223810448
Offset: 6
Original entry on oeis.org
1, 268, 31356, 2471008, 156688950, 8769288752, 457164358716, 22959079494036, 1135195898868631, 56066461542293032, 2793059847097230456, 141275485011659499936, 7287917154352619470284, 384587934464798917826784, 20802129691800205763744280, 1154767942465769017202459544
Offset: 7
Original entry on oeis.org
1, 366, 55650, 5493004, 423998498, 28236562692, 1719625254528, 99364697625603, 5581585635090040, 309848573192308938, 17191725056094651066, 960908576231662848594, 54403314773991195041214, 3132006013784614723699158, 183841318769012678798846646, 11022935912713058281592718756
Offset: 8
Original entry on oeis.org
3, 45, 637, 9567, 156123, 2781389, 54043365, 1141864959, 26137086451, 645573913005, 17138071687053, 487130581207775, 14771502665168715, 476244289169954253, 16274079501768450421, 587711952558035789055, 22370368544174736836835, 895269564785985107345453, 37585737112233443277746589
Offset: 2
-
A286781_ser(N, t='t) = {
my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
while(n++,
y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
if (y1 == y0, break()); y0 = y1; );
y0;
};
A286782_ser(N, t='t) = my(s=A286781_ser(N, t)); 1 + x*s + 2*x^2 * deriv(s,'x);
Kol(K, N=20) = {
my(v = A286782_ser(N+K+1, 't+O('t^(K+1))));
vector(N, n, polcoeff(polcoeff(v, K+n), K));
};
Kol(1)
Showing 1-8 of 8 results.
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