A124526 -id:A124526 - cp's OEIS frontend

A124527 Row sums of triangle A124526.

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

Paul D. Hanna, Nov 08 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A124526; A124528, A124529.

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))}

A124528 a(n) = Sum_{k=0..n} 2^k*A124526(n,k) for n>=0.

Original entry on oeis.org

1, 1, 3, 8, 26, 94, 377, 1639, 7623, 38034, 199338, 1111816, 6442481, 39478219, 249507483, 1659172454, 11321526022, 80944313164, 591617080305, 4514822914133, 35120998653271, 284407875530728, 2342407874087454

Offset: 0

Paul D. Hanna, Nov 08 2006

nonn

Cf. A124526, A124527, A124529.

{a(n)=sum(k=0,n\2,2^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))}

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

Paul D. Hanna, Nov 08 2006

nonn

Equals diagonal sums of triangle A124418: a(n) = Sum_{k=0..n} A124418(n+k,k) for n>=0 (conjecture).

Cf. A124526, A124527, A124528; A124418.

{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))}

A124418 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having exactly k blocks that contain both odd and even entries (0<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 9, 2, 10, 30, 12, 25, 100, 72, 6, 75, 370, 372, 60, 225, 1369, 1922, 600, 24, 780, 5587, 9920, 4500, 360, 2704, 22801, 51200, 33750, 5400, 120, 10556, 101774, 273920, 234000, 55800, 2520, 41209, 454276, 1465472, 1622400, 576600, 52920, 720

Offset: 0

Emeric Deutsch, Oct 31 2006

Row n has 1+floor(n/2) terms. Row sums are the Bell numbers (A000110). T(n,0)=A124419(n).

			T(4,1) = 9 because we have 1234, 134|2, 1|234, 124|3, 14|2|3, 1|2|34, 123|4, 1|23|4 and 12|3|4.
Triangle starts:
   1;
   1;
   1,   1;
   2,   3;
   4,   9,  2;
  10,  30, 12;
  25, 100, 72, 6;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000110, A124419, A124420, A124421, A124422, A124423.
Cf. A124526, A362495.
T(2n,n) gives A000142.

Q[0]:=1: for n from 1 to 13 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: for n from 0 to 13 do P[n]:=sort(subs({t=1,s=1},Q[n])) od: for n from 0 to 13 do seq(coeff(P[n],x,j),j=0..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
with(combinat):
T:= proc(n, k) local g, u; g:= floor(n/2); u:=ceil(n/2);
      add(binomial(g, i)*stirling2(i, k)*bell(g-i), i=k..g)*
      add(binomial(u, i)*stirling2(i, k)*bell(u-i), i=k..u)*k!
    end:
seq(seq(T(n,k), k=0..floor(n/2)), n=0..15); # Alois P. Heinz, Oct 23 2013

T[n_, k_] := Module[{g = Floor[n/2], u = Ceiling[n/2]}, Sum[Binomial[g, i] * StirlingS2[i, k]*BellB[g-i], {i, k, g}]*Sum[Binomial[u, i]*StirlingS2[i, k] * BellB[u-i], {i, k, u}]*k!]; Table[Table[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 20 2015, after Alois P. Heinz *)

{T(n,k)=if(k<0||k>n,0, 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))} \\ Paul D. Hanna, Nov 08 2006

The generating polynomial of row n is P[n](x)=Q[n](1,1,x), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.
Conjecture: T(n,k) = k!*A049020([n/2],k)*A049020([(n+1)/2],k) where A049020(n,k)=Sum_{i=0..n} S2(n,i)*C(i,k) and S2(n,k)=(1/k!)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*j^n (the Stirling numbers of 2nd kind). - Paul D. Hanna, Nov 08 2006
Sum_{k=0..floor(n/2)} = k * A362495(n). - Alois P. Heinz, Jun 05 2023

cp's OEIS Frontend

A124527 Row sums of triangle A124526.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A124528 a(n) = Sum_{k=0..n} 2^k*A124526(n,k) for n>=0.

Views

Author

Keywords

Crossrefs

Programs

PARI

A124529 a(n) = Sum_{k=0..n} k!*A124526(n+k,k) for n>=0.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A124418 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having exactly k blocks that contain both odd and even entries (0<=k<=floor(n/2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula