A187657
Binomial convolution of the central Stirling numbers of the second kind.
Original entry on oeis.org
1, 2, 16, 222, 4416, 114660, 3676814, 140408338, 6222858240, 314006546124, 17774855765140, 1115507717954432, 76871991664546170, 5770732305836768712, 468750121409142448386, 40964179307489016777630, 3832326196169482368117760
Offset: 0
-
seq(sum(binomial(n, k) *combinat[stirling2](2*k, k) *combinat[stirling2](2*(n-k), n-k), k=0..n), n=0..12);
-
Table[Sum[Binomial[n, k]StirlingS2[2k, k]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 16}]
-
makelist(sum(binomial(n,k)*stirling2(2*k,k)*stirling2(2*n-2*k, n-k),k,0,n),n,0,12);
A384471
a(n) = Sum_{k=0..n} binomial(n,k)^2 * Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k).
Original entry on oeis.org
1, 2, 18, 306, 8046, 296100, 14307254, 865996306, 63308257198, 5432272670376, 535074966419260, 59461066810476232, 7354069129792197762, 1001371912804041913056, 148806933109572134044158, 23958722845801073318076450, 4154065510530807075869275150, 771608888261061026185781127184
Offset: 0
-
Table[Sum[StirlingS2[2*k, k]*StirlingS2[2*n-2*k, n-k]*Binomial[n, k]^2, {k, 0, n}], {n, 0, 20}]
A384470
a(n) = n! * Sum_{k=0..n} Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) / binomial(n,k).
Original entry on oeis.org
1, 2, 29, 1108, 82924, 10302768, 1917699552, 499332175200, 173242955039616, 77238974345915520, 43027312823342164800, 29285800226400628915200, 23913110797474508388449280, 23071378298963178620672409600, 25964692904608781751347296204800, 33711625062334209438536728660070400
Offset: 0
-
Table[n! * Sum[StirlingS2[2*k, k] * StirlingS2[2*n-2*k, n-k] / Binomial[n, k], {k, 0, n}], {n, 0, 20}]
A384496
a(n) = Sum_{k=0..n} binomial(n,k)^3 * abs(Stirling1(2*k,k)) * abs(Stirling1(2*n-2*k,n-k)).
Original entry on oeis.org
1, 2, 30, 1044, 68474, 7180900, 1050625720, 196205015216, 44361477901818, 11751610490415828, 3567182462164189140, 1220655384720089761080, 464932034143270233958352, 195108754505934104188716064, 89452431045403310104416682304, 44489455448017524780072427344000
Offset: 0
-
Table[Sum[Binomial[n,k]^3 * Abs[StirlingS1[2*k,k]] * Abs[StirlingS1[2*n-2*k,n-k]], {k, 0, n}], {n, 0, 20}]
A384491
a(n) = n!^2 * Sum_{k=0..n} Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) / binomial(n,k)^2.
Original entry on oeis.org
1, 2, 57, 6536, 1966816, 1226860992, 1373652478656, 2507498281198080, 6966291361870181376, 27969794062091821670400, 155875927262331497576140800, 1167389777699203314381963264000, 11441270265465265986005655905894400, 143525982910350708912088976768630784000
Offset: 0
-
Table[n!^2 * Sum[StirlingS2[2*k, k] * StirlingS2[2*n-2*k, n-k] / Binomial[n, k]^2, {k, 0, n}], {n, 0, 15}]
-
a(n) = n!^2 * sum(k=0, n, stirling(2*k,k, 2) * stirling(2*n-2*k,n-k,2) / binomial(n,k)^2); \\ Michel Marcus, May 31 2025
A384492
a(n) = n!^3 * Sum_{k=0..n} Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) / binomial(n,k)^3.
Original entry on oeis.org
1, 2, 113, 38992, 47071264, 147015606528, 988250901343488, 12631667044878213120, 280790763724247161061376, 10147405862241529912885248000, 565550513462476798468573003776000, 46592777163703224212146175606784000000, 5479872142880875751798643810680954683392000
Offset: 0
-
Table[n!^3 * Sum[StirlingS2[2*k, k] * StirlingS2[2*n-2*k, n-k] / Binomial[n, k]^3, {k, 0, n}], {n, 0, 15}]
-
a(n) = n!^3 * sum(k=0, n, stirling(2*k,k,2) * stirling(2*n-2*k,n-k,2) / binomial(n,k)^3); \\ Michel Marcus, May 31 2025
Showing 1-6 of 6 results.
Comments