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A124424 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n}, having exactly k blocks consisting of entries of the same parity (0<=k<=n). 2
1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 3, 4, 5, 2, 1, 7, 14, 16, 10, 4, 1, 25, 48, 61, 42, 20, 6, 1, 79, 194, 250, 200, 106, 38, 9, 1, 339, 820, 1145, 958, 569, 230, 66, 12, 1, 1351, 3794, 5554, 5096, 3251, 1486, 486, 112, 16, 1, 6721, 18960, 29101, 28010, 19110, 9470, 3477, 930, 175, 20, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Row sums are the Bell numbers (A000110). T(n,0)=A124425(n).

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

The generating polynomial of row n is P[n](t)=Q[n](t,t,1), 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.

EXAMPLE

T(4,2) = 5 because we have 13|24, 14|2|3, 1|2|34, 1|23|4 and 12|3|4.

Triangle starts:

1;

0,  1;

1,  0,  1;

1,  2,  1,  1;

3,  4,  5,  2, 1;

7, 14, 16, 10, 4, 1;

MAPLE

Q[0]:=1: for n from 1 to 11 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 11 do P[n]:=sort(subs({s=t, x=1}, Q[n])) od: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form

# second Maple program:

b:= proc(g, u) option remember;

      add(Stirling2(g, k)*Stirling2(u, k)*k!, k=0..min(g, u))

    end:

T:= proc(n, k) local g, u; g:= floor(n/2); u:= ceil(n/2);

      add(add(add(binomial(g, i)*Stirling2(i, h)*binomial(u, j)*

      Stirling2(j, k-h)*b(g-i, u-j), j=k-h..u), i=h..g), h=0..k)

    end:

seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Oct 24 2013

MATHEMATICA

b[g_, u_] := b[g, u] = Sum[StirlingS2[g, k]*StirlingS2[u, k]*k!, {k, 0, Min[g, u]}] ; T[n_, k_] := Module[{g, u}, g = Floor[n/2]; u = Ceiling[n/2]; Sum[ Sum[ Sum[ Binomial[g, i]*StirlingS2[i, h]*Binomial[u, j]*StirlingS2[j, k-h]*b[g-i, u-j], {j, k-h, u}], {i, h, g}], {h, 0, k}]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Feb 19 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A000110, A124418, A124419, A124420, A124421, A124422, A124423, A124425.

Sequence in context: A210691 A262175 A278028 * A057044 A153899 A068098

Adjacent sequences:  A124421 A124422 A124423 * A124425 A124426 A124427

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Nov 01 2006

STATUS

approved

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Last modified January 20 21:36 EST 2019. Contains 319336 sequences. (Running on oeis4.)