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A291844 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Row n>0 contains floor((2*n+2)/3) terms.
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.
FORMULA
y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies y = ((1+x)*z - 1) * (1 + t*x)/((1-t + t*(1+x)*z)*x*z^2), where z = A291843(x;t) and P_n(t) = Sum_{k=0..floor((2*n-1)/3)} T(n,k)*t^k for n > 0.
A294158(n) = P_n(1), A294159(n)=P_n(-1), A294160(n)=P_n(0).
EXAMPLE
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] ...
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 *)
PROG
(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))))
CROSSREFS
Columns k=0..5 give A294160 (k=0), A294161 (k=1), A294162 (k=2), A294163 (k=3), A294164 (k=4), A294165 (k=5).
Sequence in context: A200032 A121667 A368767 * A353750 A093991 A030447
KEYWORD
nonn,tabf
AUTHOR
Gheorghe Coserea, Oct 24 2017
STATUS
approved

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)