A291844 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
1, 1, 4, 2, 29, 23, 274, 292, 36, 3145, 4068, 994, 16, 42294, 62861, 22250, 1512, 651227, 1075562, 484840, 61027, 1060, 11295242, 20275944, 10867381, 1977879, 93188, 280, 217954807, 418724047, 255929070, 59896915, 4823178, 80632, 4632600152, 9418874022, 6387031115, 1798212190, 204846125, 7410676, 37056, 107572674851, 229535650138, 169414005231, 55017177704, 8022471066, 463514918, 7255380, 7040
Offset: 0
Examples
A(x;t) = 1 + x + (4 + 2*t)*x^2 + (29 + 23*t)*x^3 + (274 + 292*t + 36*t^2)*x^4 + ... Triangle starts: n\k [0] [1] [2] [3] [4] [5] [0] 1; [1] 1; [2] 4, 2; [3] 29, 23; [4] 274, 292, 36; [5] 3145, 4068, 994, 16; [6] 42294, 62861, 22250, 1512; [7] 651227, 1075562, 484840, 61027, 1060; [8] 11295242, 20275944, 10867381, 1977879, 93188, 280; [9] 217954807, 418724047, 255929070, 59896915, 4823178, 80632; [10] ...
Links
- Gheorghe Coserea, Rows n = 0..124, flattened
- Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.
Crossrefs
Programs
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Mathematica
m = maxExponent = 13; Z[_] = 0; Do[Z[t_] = -(((1 - l + l (1+t) Z[t]) (-((t Z[t])/(1 + l t)) - (1 - t - 2 l t^2)/(1 - l + l (1+t) Z[t]) - 2 t^2 Z'[t]))/((1+t) (1 - t - 2 l t^2))) + O[t]^m // Normal // Simplify, {m}]; gamma[t_] = ((1 + l t)(-1 + Z[t] + t Z[t]))/(Z[t]^2 (t + l t (-1 + Z[t] + t Z[t]))) + O[t]^m // Normal // Simplify; CoefficientList[# + O[l]^m, l]& /@ Most @ CoefficientList[gamma[t], t] // Flatten (* Jean-François Alcover, Nov 17 2018 *) -
PARI
A291843_ser(N, t='t) = { my(x='x+O('x^N), y=1, y1=0, n=1, dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1)); while (n++, y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) + (t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn; if (y1 == y, break); y = y1;); y; }; A291844_ser(N, t='t) = { my(z = A291843_ser(N+1,t)); ((1+x)*z - 1)*(1 + t*x)/((1-t + t*(1+x)*z)*x*z^2); }; concat(apply(p->Vecrev(p), Vec(A291844_ser(12))))
Comments