A383818
Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - k*x) * Product_{j=0..k-1} (1 + j*x)/(1 - j*x).
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 9, 10, 1, 0, 1, 16, 45, 22, 1, 0, 1, 25, 136, 177, 46, 1, 0, 1, 36, 325, 856, 621, 94, 1, 0, 1, 49, 666, 3025, 4576, 2049, 190, 1, 0, 1, 64, 1225, 8646, 23125, 22216, 6525, 382, 1, 0, 1, 81, 2080, 21217, 90126, 156145, 101536, 20337, 766, 1, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 4, 9, 16, 25, ...
0, 1, 10, 45, 136, 325, ...
0, 1, 22, 177, 856, 3025, ...
0, 1, 46, 621, 4576, 23125, ...
0, 1, 94, 2049, 22216, 156145, ...
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a(n, k) = sum(j=0, k, abs(stirling(k, j, 1))*stirling(j+n, k, 2));
A383843
Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/Product_{j=0..k} (1 - j*x)^2.
Original entry on oeis.org
1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 23, 4, 0, 1, 20, 86, 72, 5, 0, 1, 30, 230, 480, 201, 6, 0, 1, 42, 505, 2000, 2307, 522, 7, 0, 1, 56, 973, 6300, 14627, 10044, 1291, 8, 0, 1, 72, 1708, 16464, 65002, 95060, 40792, 3084, 9, 0, 1, 90, 2796, 37632, 227542, 587580, 567240, 157440, 7181, 10, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 2, 6, 12, 20, 30, 42, ...
0, 3, 23, 86, 230, 505, 973, ...
0, 4, 72, 480, 2000, 6300, 16464, ...
0, 5, 201, 2307, 14627, 65002, 227542, ...
0, 6, 522, 10044, 95060, 587580, 2725380, ...
0, 7, 1291, 40792, 567240, 4817990, 29331038, ...
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a(n, k) = sum(j=0, n, stirling(j+k, k, 2)*stirling(n-j+k, k, 2));
A383883
a(n) = [x^n] 1/((1 - n*x) * Product_{k=0..n-1} (1 - k*x)^2).
Original entry on oeis.org
1, 1, 11, 222, 6627, 262570, 12978758, 769079444, 53138842515, 4194648739710, 372421403333850, 36733739199892020, 3985122473105099406, 471598870326072262644, 60456151456891375730860, 8345905345383943433713800, 1234395864446065862689721475, 194738649118647202909304657910
Offset: 0
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Join[{1}, Table[Sum[StirlingS2[n + k - 1, n - 1]*StirlingS2[2*n - k, n], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 14 2025 *)
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a(n) = polcoef(1/((1-n*x)*prod(k=0, n-1, 1-k*x+x*O(x^n))^2), n);
A383892
Expansion of 1/( ((1-x)*(1-2*x)*(1-3*x)*(1-4*x))^2 * (1-5*x) ).
Original entry on oeis.org
1, 25, 355, 3775, 33502, 262570, 1880090, 12574850, 79778303, 485441135, 2856558005, 16358449625, 91615095204, 503740623720, 2727832278900, 14584759018500, 77152991893005, 404503014170325, 2104862289863575, 10883633564375875, 55976319375728506, 286601257317512950
Offset: 0
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- Index entries for linear recurrences with constant coefficients, signature (25,-270,1650,-6273,15345,-24080,23300,-12576,2880).
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[1] cat [&+[StirlingSecond(k+4,4) * StirlingSecond(n-k+5,5): k in [0..n]]: n in [1..25]]; // Vincenzo Librandi, May 23 2025
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a[n_]:=Sum [StirlingS2[k+4,4]*StirlingS2[n-k+5,5],{k,0,n}];Table[a[n],{n,0,19}] (* Vincenzo Librandi, May 23 2025 *)
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a(n) = sum(k=0, n, stirling(k+4, 4, 2)*stirling(n-k+5, 5, 2));
A383893
Expansion of 1/( ((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x))^2 * (1-6*x) ).
Original entry on oeis.org
1, 36, 721, 10626, 128758, 1360128, 12978758, 114537348, 950326391, 7502910996, 56878787231, 416937779286, 2971567050420, 20682844799760, 141092113563660, 946112664225960, 6251628891468765, 40789040893547940, 263235445374827965, 1682802305881045290, 10669738322822387746
Offset: 0
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- Index entries for linear recurrences with constant coefficients, signature (36,-575,5370,-32523,133848,-381065,748530,-991276,840216,-408960,86400).
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[1] cat [&+[StirlingSecond(k+5,5) * StirlingSecond(n-k+6,6): k in [0..n]]: n in [1..25]]; // Vincenzo Librandi, May 23 2025
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a[n_]:=Sum [StirlingS2[k+5,5]*StirlingS2[n-k+6,6],{k,0,n}];Table[a[n],{n,0,19}] (* Vincenzo Librandi, May 23 2025 *)
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a(n) = sum(k=0, n, stirling(k+5, 5, 2)*stirling(n-k+6, 6, 2));
Showing 1-5 of 5 results.