A132007
Triangle, read by rows, where T(n,k) = T(n,k-1) + n*T(n-1,k-1) for n>0 and k>0, with T(n,0) = T(n-1,n-1) for n>0 and T(0,0) = 1.
Original entry on oeis.org
1, 1, 2, 2, 4, 8, 8, 14, 26, 50, 50, 82, 138, 242, 442, 442, 692, 1102, 1792, 3002, 5212, 5212, 7864, 12016, 18628, 29380, 47392, 78664, 78664, 115148, 170196, 254308, 384704, 590364, 922108, 1472756, 1472756, 2102068, 3023252, 4384820, 6419284
Offset: 0
Triangle begins:
1;
1, 2;
2, 4, 8;
8, 14, 26, 50;
50, 82, 138, 242, 442;
442, 692, 1102, 1792, 3002, 5212;
5212, 7864, 12016, 18628, 29380, 47392, 78664;
78664, 115148, 170196, 254308, 384704, 590364, 922108, 1472756;
1472756, 2102068, 3023252, 4384820, 6419284, 9496916, 14219828, 21596692, 33378740; ...
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T[n_, k_] := T[n, k] = If[k < 0 || n < k, 0, If[n == 0 && k == 0, 1, If[k == 0, T[n - 1, n - 1], T[n, k - 1] + n*T[n - 1, k - 1]]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 15 2017 *)
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T(n,k)=if(k<0 || n
Original entry on oeis.org
1, 3, 14, 98, 954, 12242, 199156, 3988248, 96094356, 2735515916, 90647239704, 3453539346232, 149695821098056, 7316061672718152, 400012182207670288, 24301065118906010048, 1630456684740565109904, 120168318315515773499952
Offset: 0
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T[n_, k_] := T[n, k] = If[k < 0 || n < k, 0, If[n == 0 && k == 0, 1, If[k == 0, T[n - 1, n - 1], T[n, k - 1] + n*T[n - 1, k - 1]]]]; Table[Sum[T[n, k], {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Dec 15 2017 *)
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{ T(n, k)=if(k<0 || nG. C. Greubel, Dec 15 2017
A331660
E.g.f. A(x) satisfies: d/dx A(x) = 1 + (1/(1 - x)) * A(x/(1 - x)).
Original entry on oeis.org
1, 1, 5, 32, 280, 3280, 49480, 927560, 21037640, 566134160, 17803754560, 646052181520, 26757321804880, 1252934215973600, 65791336312915520, 3846554938702140320, 248841434876849499040, 17713758333248102781760, 1380631354206969100115200
Offset: 1
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terms = 20; A[] = 0; Do[A[x] = Normal[Integrate[1 + 1/(1 - x) A[x/(1 - x) + O[x]^(terms + 1)], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]! // Rest
a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k]^2 k! a[n - k - 1], {k, 0, n - 2}]; Table[a[n], {n, 1, 20}]
A336243
a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * (n-k-1)! * a(k).
Original entry on oeis.org
1, 1, 5, 56, 1114, 34624, 1549648, 94402356, 7511324448, 756406501200, 94039208461584, 14146468841290752, 2532586289913605088, 532113978869395649856, 129662518122880634567232, 36270261084908437106586624, 11543682123659880166705099776
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[1/(1 - Sum[x^k/(k k!), {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2
nmax = 16; Assuming[x > 0, CoefficientList[Series[1/(1 + EulerGamma - ExpIntegralEi[x] + Log[x]), {x, 0, nmax}], x]] Range[0, nmax]!^2
Showing 1-4 of 4 results.
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