A124526
Triangle, read by rows, where T(n,k) = A049020([n/2],k)*A049020([(n+1)/2],k).
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 4, 9, 1, 10, 30, 6, 25, 100, 36, 1, 75, 370, 186, 10, 225, 1369, 961, 100, 1, 780, 5587, 4960, 750, 15, 2704, 22801, 25600, 5625, 225, 1, 10556, 101774, 136960, 39000, 2325, 21, 41209, 454276, 732736, 270400, 24025, 441, 1, 178031, 2199262, 4110512, 1849120, 217000, 6027, 28, 769129, 10647169, 23059204, 12645136, 1960000, 82369, 784, 1, 3630780, 55493841, 136074274, 87570056, 16787400, 944230, 13720, 36
Offset: 0
Triangle begins:
1;
1;
1, 1;
2, 3;
4, 9, 1;
10, 30, 6;
25, 100, 36, 1;
75, 370, 186, 10;
225, 1369, 961, 100, 1;
780, 5587, 4960, 750, 15;
2704, 22801, 25600, 5625, 225, 1;
10556, 101774, 136960, 39000, 2325, 21;
41209, 454276, 732736, 270400, 24025, 441, 1;
178031, 2199262, 4110512, 1849120, 217000, 6027, 28;
769129, 10647169, 23059204, 12645136, 1960000, 82369, 784, 1;
3630780, 55493841, 136074274, 87570056, 16787400, 944230, 13720, 36; ...
-
S[n_, k_] = Sum[StirlingS2[n, i] Binomial[i, k], {i, 0, n}];
T[n_, k_] := S[Floor[n/2], k] S[Floor[(n+1)/2], k];
Table[T[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Nov 02 2020 *)
-
{T(n,k) = (n\2)!*((n+1)\2)!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),n\2),k) *polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),(n+1)\2),k)}
for(n=0,15, for(k=0,n\2, print1(T(n,k),", "));print(""))
Original entry on oeis.org
1, 1, 2, 5, 14, 46, 162, 641, 2656, 12092, 56956, 290636, 1523088, 8559980, 49163792, 300514337, 1870652672, 12318376190, 82394305842, 580168452664, 4141242464512, 30992978322024, 234765130286990, 1858132080028884
Offset: 0
-
b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
`if`(n=0, 1, b(n-1, k-1) +(k+1)*(b(n-1, k) +b(n-1, k+1))))
end:
a:= n-> add(b(iquo(n, 2), k)*b(iquo(n+1, 2), k), k=0..n/2):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 14 2014
-
b[n_, k_] := b[n, k] = If[k < 0 || k > n, 0, If[n == 0, 1, b[n - 1, k - 1] + (k + 1) (b[n - 1, k] + b[n - 1, k + 1])]];
a[n_] := Sum[b[Quotient[n, 2], k] b[Quotient[n + 1, 2], k], {k, 0, n/2}];
a /@ Range[0, 30]
(* Second program: *)
S[n_, k_] = Sum[StirlingS2[n, i] Binomial[i, k], {i, 0, n}];
T[n_, k_] := S[Floor[n/2], k] S[Floor[(n + 1)/2], k];
a[n_] := Sum[T[n, k], {k, 0, Floor[n/2]}];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 02 2020, first program after Alois P. Heinz *)
-
{a(n)=sum(k=0,n\2,(n\2)!*((n+1)\2)!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),n\2),k) *polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),(n+1)\2),k))}
A124529
a(n) = Sum_{k=0..n} k!*A124526(n+k,k) for n>=0.
Original entry on oeis.org
1, 2, 6, 29, 190, 1562, 15457, 179034, 2377092, 35599701, 593731310, 10914169312, 219252994039, 4779086510108, 112341582757512, 2833025331800643, 76293601822430388, 2185288262904326236, 66338823231846583471
Offset: 0
-
{a(n)=sum(k=0,n,k!*((n+k)\2)!*((n+k+1)\2)!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),(n+k)\2),k) *polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),(n+k+1)\2),k))}
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