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A264398 Triangle read by rows: T(n,k) is the number of partitions of n having k parts with odd multiplicities. 3
1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 1, 0, 4, 3, 3, 4, 3, 1, 0, 7, 7, 1, 5, 7, 7, 3, 0, 12, 14, 4, 7, 12, 14, 8, 1, 0, 19, 26, 10, 1, 11, 19, 26, 18, 3, 0, 30, 45, 22, 4, 15, 30, 45, 36, 9, 0, 45, 75, 44, 11, 1, 22, 45, 75, 67, 21, 1, 0, 67, 120, 81, 26, 3, 30, 67, 120, 119, 45, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

T(n,0) = A035363(n).

Row sums give A000041. - Omar E. Pol, Nov 21 2015

LINKS

Alois P. Heinz, Rows n = 0..1000, flattened

FORMULA

G.f.: G(t,x) = Product_{j>=1} (1 + tx^j)/(1 - x^(2j)).

EXAMPLE

T(6,1) = 4 because we have [6*], [4*,1,1],[2*,2,2], and [2*,1,1,1,1] (parts with odd multiplicities are marked).

Triangle starts:

1;

0,1;

1,1;

0,2,1;

2,2,1;

0,4,3;

3,4,3,1;

MAPLE

g := product((1+t*x^j)/(1-x^(2*j)), j = 1 .. 100): gser := simplify(series(g, x = 0, 30)): for n from 0 to 28 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 23 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form

# second Maple program:

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(expand(`if`(j::odd, x, 1)*b(n-i*j, i-1)), j=0..n/i)))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):

seq(T(n), n=0..20);  # Alois P. Heinz, Nov 25 2015

MATHEMATICA

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Expand[If[OddQ[j], x, 1]* b[n-i*j, i-1]], {j, 0, n/i}]]]; T[n_] := Function[p, Table[Coefficient[ p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-Fran├žois Alcover, Feb 18 2016, after Alois P. Heinz *)

PROG

(PARI)

T(n) = { Vec(prod(k=1, n, (1 + y*x^k)/(1 - x^(2*k)) + O(x*x^n))) }

{ my(t=T(10)); for(n=1, #t, print(Vecrev(t[n]))); } \\ Andrew Howroyd, Dec 22 2017

CROSSREFS

Cf. A000041, A035363, A264399, A264400.

Sequence in context: A178064 A145363 A071429 * A145364 A284978 A175862

Adjacent sequences:  A264395 A264396 A264397 * A264399 A264400 A264401

KEYWORD

nonn,look,tabf

AUTHOR

Emeric Deutsch, Nov 21 2015

STATUS

approved

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Last modified July 23 03:00 EDT 2019. Contains 325230 sequences. (Running on oeis4.)