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A101477
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Square array T(n,k), read by antidiagonals: number of labeled trees, with increments of labels along edges constrained to +-1, with n nodes that have no label greater than k.
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1
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1, 1, 1, 1, 2, 3, 1, 2, 7, 12, 1, 2, 8, 31, 56, 1, 2, 8, 39, 156, 288, 1, 2, 8, 40, 211, 851, 1584, 1, 2, 8, 40, 223, 1219, 4909, 9152, 1, 2, 8, 40, 224, 1327, 7371, 29506, 54912, 1, 2, 8, 40, 224, 1343, 8250, 46099, 183043, 339456, 1, 2, 8, 40, 224, 1344, 8427, 52938, 295915, 1164387, 2149888
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OFFSET
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0,5
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LINKS
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FORMULA
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G.f. of k-th row: A(t)=B(t)*(1-C(t)^(k+1))*(1-C(t)^(k+5))/[(1-C(t)^(k+2))*(1-C(t)^(k+4))], with tB(t) the g.f. of A052701 and C(t) the g.f. of A101478.
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EXAMPLE
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1, 1, 3, 12, 56, 288, 1584, 9152, 54912, 339456, ...
1, 2, 7, 31, 156, 851, 4909, 29506, 183043, 1164387, ...
1, 2, 8, 39, 211, 1219, 7371, 46099, 295915, 1939395, ...
1, 2, 8, 40, 223, 1327, 8250, 52938, 347941, 2330532, ...
1, 2, 8, 40, 224, 1343, 8427, 54625, 362833, 2456261, ...
1, 2, 8, 40, 224, 1344, 8447, 54887, 365688, 2484384, ...
1, 2, 8, 40, 224, 1344, 8448, 54911, 366051, 2488831, ...
1, 2, 8, 40, 224, 1344, 8448, 54912, 366079, 2489311, ...
1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489343, ...
1, 2, 8, 40, 224, 1344, 8448, 54912, 366080, 2489344, ...
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MATHEMATICA
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nmax = 11;
b[x_] = Sum[2^(n - 1)*(2*n - 2)!/(n - 1)!/n! x^n, {n, 1, nmax}];
c[x_] = 0; Do[c[x_] = x*(1 + c[x])^4/(1 + c[x]^2) + O[x]^nmax, {nmax}];
a[n_, t_] := a[n, t] = b[t]*(1 - c[t]^(n + 1))*(1 - c[t]^(n + 5))/((1 - c[t]^(n + 2))*(1 - c[t]^(n + 4)));
T[n_, k_] := SeriesCoefficient[a[n, t], {t, 0, k}];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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