A287030
Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
Original entry on oeis.org
0, 0, 1, 1, 1, 2, 1, 1, 3, 5, 3, 9, 17, 6, 2, 30, 41, 26, 10, 81, 131, 111, 30, 5, 242, 491, 357, 134, 35, 838, 1625, 1274, 652, 140, 14, 2799, 5497, 5202, 2556, 676, 126, 9365, 20581, 19827, 10200, 3610, 630, 42, 33616, 76561, 74797, 44880, 16390, 3334, 462, 122937, 282591, 301188, 190278, 72490, 19218, 2772, 132, 449698, 1089375, 1219920, 788654, 341770, 97890, 16108, 1716, 1696724, 4285737, 4893603, 3398950, 1578577, 474838, 99386, 12012, 429
Offset: 0
A(x;t) = t*x + (1 + t)*x^2 + (2 + t + t^2)*x^3 + (3 + 5*t + 3*t^2)*x^4 + (9 + 17*t + 6*t^2 + 2*t^3)*x^5 + ...
Triangle starts:
n\k [0] [1] [2] [3] [4] [5] [6] [7]
[0] 0;
[1] 0, 1;
[2] 1, 1;
[3] 2, 1, 1;
[4] 3, 5, 3;
[5] 9, 17, 6, 2;
[6] 30, 41, 26, 10;
[7] 81, 131, 111, 30, 5;
[8] 242, 491, 357, 134, 35;
[9] 838, 1625, 1274, 652, 140, 14;
[10] 2799, 5497, 5202, 2556, 676, 126;
[11] 9365, 20581, 19827, 10200, 3610, 630, 42;
[12] 33616, 76561, 74797, 44880, 16390, 3334, 462;
[13] 122937, 282591, 301188, 190278, 72490, 19218, 2772, 132;
[14] 449698, 1089375, 1219920, 788654, 341770, 97890, 16108, 1716;
[15] ...
-
max = 15; y[, ] = 0;
Do[y[x_, t_] = Series[t x + x y[x, t]^2 + x D[y[x, t], t] + x y[x, t], {x, 0, max}, {t, 0, max}] // Normal, max];
CoefficientList[#, t]& /@ CoefficientList[y[x, t], x] /. {} -> {0} // Flatten (* Jean-François Alcover, Oct 25 2018 *)
-
A287030_ser(N) = {
my(x='x+O('x^N), F0=x, t='t, F1=0, n=1);
while(n++,
F1 = t*x + x*F0^2 + x*deriv(F0,t) + x*F0;
if (F1 == F0, break()); F0 = F1;); F0;
};
concat(0, concat(apply(p->Vecrev(p), Vec(A287030_ser(16)))))
\\ test: y=A287030_ser(100); y == t*x + x*y^2 + x*deriv(y,t) + x*y
A287040
Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
Original entry on oeis.org
0, 1, 1, 1, 1, 2, 5, 3, 2, 8, 17, 22, 10, 5, 29, 91, 106, 94, 35, 14, 140, 431, 701, 582, 396, 126, 42, 661, 2501, 4067, 4544, 2980, 1654, 462, 132, 3622, 14025, 27394, 31032, 26680, 14598, 6868, 1716, 429, 19993, 87947, 177018, 236940, 208780, 146862, 69356, 28396, 6435, 1430, 120909, 550811, 1245517, 1727148, 1776310, 1291654, 772422, 322204, 117016, 24310, 4862
Offset: 0
A(x;t) = t + (1 + t + t^2)*x + (2 + 5*t + 3*t^2 + 2*t^3)*x^2 + (8 + 17*t + 22*t^2 + 10*t^3 + 5*t^4)*x^3 + ...
Triangle starts:
n\k [0] [1] [2] [3] [4] [5] [6] [7] [8] [9]
[0] 0, 1;
[1] 1, 1, 1;
[2] 2, 5, 3, 2;
[3] 8, 17, 22, 10, 5;
[4] 29, 91, 106, 94, 35, 14;
[5] 140, 431, 701, 582, 396, 126, 42;
[6] 661, 2501, 4067, 4544, 2980, 1654, 462, 132;
[7] 3622, 14025, 27394, 31032, 26680, 14598, 6868, 1716, 429;
[8] 19993, 87947, 177018, 236940, 208780, 146862, 69356, 28396, 6435, 1430;
[9] ...
-
nmax = 10; y[0, t_] := t; y[, ] = 0;
Do[y[x_, t_] = Series[t + x y[x, t]^2 + x D[y[x, t], t] + x y[x, t], {x, 0, nmax}, {t, 0, nmax}] // Normal, {n, 0, nmax}];
CoefficientList[#, t]& /@ CoefficientList[y[x, t]+O[x]^nmax, x] // Flatten (* Jean-François Alcover, Dec 13 2018 *)
-
A287040_ser(N) = {
my(x='x+O('x^N), t='t, F0=t, F1=0, n=1);
while(n++,
F1 = t + x*F0^2 + x*deriv(F0, t) + x*F0;
if (F1 == F0, break()); F0 = F1; ); F0;
};
concat(apply(p->Vecrev(p), Vec(A287040_ser(10))))
\\ test: y=A287040_ser(50); y == t + x*y^2 + x*deriv(y, t) + x*y
A287045
a(n) is the number of size n affine closed terms of variable size 0.
Original entry on oeis.org
0, 1, 2, 8, 29, 140, 661, 3622, 19993, 120909, 744890, 4887401, 32795272, 230728608, 1661537689, 12426619200, 95087157771, 750968991327, 6062088334528, 50288003979444, 425889463252945, 3694698371069796, 32683415513480237, 295430131502604353, 2719833636188015674, 25536232370225996575
Offset: 0
A(x) = x + 2*x^2 + 8*x^3 + 29*x^4 + 140*x^5 + ...
-
a[n_] := a[n] = If[n<3, n, (3a[n-1] + (6n-10) a[n-2] - a[n-3] + 2b[n-1] - b[n-2] - b[n-3])/2]; b[n_] := Sum[a[k] a[n-k], {k, 1, n-1}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 13 2018 *)
-
A287040_ser(N) = {
my(x='x+O('x^N), t='t, F0=t, F1=0, n=1);
while(n++,
F1 = t + x*F0^2 + x*deriv(F0, t) + x*F0;
if (F1 == F0, break()); F0 = F1; ); F0;
};
concat(0, Vec(subst(A287040_ser(26), 't, 0)))
-
A287045_seq(N) = {
my(a = vector(N), b=vector(N), t1=0);
a[1]=1; a[2]=2; a[3]=8; b[1]=0; b[2]=1; b[3]=4;
for (n=4, N, b[n] = sum(k=1, n-1, a[k]*a[n-k]);
t1 = 3*a[n-1] + (6*n-10)*a[n-2] - a[n-3];
a[n] = (t1 + 2*b[n-1] - b[n-2] - b[n-3])/2);
concat(0,a);
};
A287045_seq(25)
\\ test: y=Ser(A287045_seq(200)); 0 == 6*x^3*y' - x*(x-1)*(x+2)*y^2 - (x^3-2*x^2-3*x+2)*y + x^2 + 2*x
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