A354253
Expansion of e.g.f. 1/sqrt(9 - 8 * exp(x)).
Original entry on oeis.org
1, 4, 52, 1108, 32980, 1261204, 58928212, 3253363348, 207225008980, 14958174725524, 1206698072485972, 107589343503498388, 10505997552329149780, 1115087729794287434644, 127819745001180490920532, 15736779719362919373550228, 2071062794354825889656471380
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(9-8*exp(x))))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(2*(exp(x)-1))^k)))
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a(n) = sum(k=0, n, 2^k*(2*k)!*stirling(n, k, 2)/k!);
A292783
Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/sqrt(1 - 2*k*x).
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 12, 15, 0, 1, 4, 27, 120, 105, 0, 1, 5, 48, 405, 1680, 945, 0, 1, 6, 75, 960, 8505, 30240, 10395, 0, 1, 7, 108, 1875, 26880, 229635, 665280, 135135, 0, 1, 8, 147, 3240, 65625, 967680, 7577955, 17297280, 2027025, 0, 1, 9, 192, 5145, 136080, 2953125, 42577920, 295540245, 518918400, 34459425, 0
Offset: 0
E.g.f. of column k: A_k(x) = 1 + k*x/1! + 3*k^2*x^2/2! + 15*k^3*x^3/3! + 105*k^4*x^4/4! + 945*k^5*x^5/5! + 10395*k^6*x^6/6! +
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 3, 12, 27, 48, 75, ...
0, 15, 120, 405, 960, 1875, ...
0, 105, 1680, 8505, 26880, 65625, ...
0, 945, 30240, 229635, 967680, 2953125, ...
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Table[Function[k, n! SeriesCoefficient[1/Sqrt[1 - 2 k x], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-i k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
A345103
a(n) = 1 + 4 * Sum_{k=0..n-1} binomial(n,k) * a(k) * a(n-k-1).
Original entry on oeis.org
1, 5, 61, 1277, 37741, 1437725, 67013101, 3693540317, 234974905261, 16945434018845, 1366008048556141, 121721015465713757, 11880107754103150381, 1260413749895624939165, 144427420001275864755181, 17776090894283922227621597, 2338833689096321086977341101, 327585830473259220341296486685
Offset: 0
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a[n_] := a[n] = 1 + 4 Sum[Binomial[n, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[x]/Sqrt[9 - 8 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[Binomial[n, k] StirlingS2[k, j] 4^j (2 j - 1)!!, {j, 0, k}], {k, 0, n}], {n, 0, 17}]
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N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/sqrt(9-8*exp(x)))) \\ Seiichi Manyama, Oct 20 2021
A156654
Triangle T(n, k) = coefficients of p(x,n), where p(x,n) = ((1-x)^(2*n+1)/x^n) * Sum_{j >= n} ( (2*j+1)^n * binomial(j, n) * x^j ), read by rows.
Original entry on oeis.org
1, 3, 1, 25, 22, 1, 343, 515, 101, 1, 6561, 14156, 5766, 396, 1, 161051, 456197, 299342, 49642, 1447, 1, 4826809, 16985858, 15796159, 4592764, 371239, 5090, 1, 170859375, 719818759, 878976219, 383355555, 58474285, 2550165, 17481, 1, 6975757441, 34264190872, 52246537948, 31191262504, 7488334150, 660394024, 16574428, 59032, 1
Offset: 0
Triangle begins as:
1;
3, 1;
25, 22, 1;
343, 515, 101, 1;
6561, 14156, 5766, 396, 1;
161051, 456197, 299342, 49642, 1447, 1;
4826809, 16985858, 15796159, 4592764, 371239, 5090, 1;
170859375, 719818759, 878976219, 383355555, 58474285, 2550165, 17481, 1;
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m:= 40; R:=PowerSeriesRing(Rationals(), m);
T:= func< n | Coefficients(R!( ((1-x)^(2*n+1)/x^n)*(&+[ (2*j+1)^n*Binomial(j, n)*x^j: j in [n..m]] ) )) >;
[T(n): n in [0..12]]; // G. C. Greubel, Apr 02 2021
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p[x_, n_]:= ((1-x)^(2*n+1)/x^n)*Sum[(2*j+1)^n*Binomial[j, n]*x^j, {j,n,2*n}];
Table[CoefficientList[Series[p[x,n], {x,0,n}], x], {n,0,12}]//Flatten (* modified by G. C. Greubel, Apr 02 2021 *)
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def p(n, x): return ((1-x)^(2*n+1)/x^n)*sum( (2*j+1)^n*binomial(j, n)*x^j for j in (n..2*n) )
flatten([[( p(n, x) ).series(x, n+1).list()[k] for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 02 2021
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