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A103919 Triangle of numbers of partitions of n with total number of odd parts equal to k from {0,...,n}. 5
1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 1, 0, 4, 0, 2, 0, 1, 3, 0, 5, 0, 2, 0, 1, 0, 7, 0, 5, 0, 2, 0, 1, 5, 0, 9, 0, 5, 0, 2, 0, 1, 0, 12, 0, 10, 0, 5, 0, 2, 0, 1, 7, 0, 17, 0, 10, 0, 5, 0, 2, 0, 1, 0, 19, 0, 19, 0, 10, 0, 5, 0, 2, 0, 1, 11, 0, 28, 0, 20, 0, 10, 0, 5, 0, 2, 0, 1, 0, 30, 0, 33, 0, 20, 0, 10, 0, 5, 0, 2, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

The partition (0) of n=0 is included. For n>0 no part 0 appears.

The first (k=0) column gives the number of partitions without odd parts, i.e., those with even parts only. See A035363.

Without the alternating zeros this becomes a triangle with columns given by the rows of the S_n(m) table shown in the Riordan reference.

From Gregory L. Simay, Oct 31 2015: (Start)

T(2n+k,k) = the number of partitions of n with parts 1..k of two kinds. If n<=k, then T(2n+k) = A000712(n), the number of partitions of n with parts of two kinds.

T(2n+k) = the convolution of A000041(n) and the number of partitions of n+k having exactly k parts.

T(2n+k) = d(n,k) where d(n,0) = p(n) and d(n,k) = d(n,k-1) + d(n-k,k-1) + d(n-2k,k-1) + ... (End)

From Emeric Deutsch, Oct 04 2016: (Start)

T(n,k) = number of partitions (p1 >= p2 >= p3 >= ...) of n having alternating sum p1 - p2 + p3 - ... = k. Example: T(5,3) = 2 because there are two partitions (3,1,1) and (4,1) of 5 with alternating sum 3.

The equidistribution of the partition statistics "alternating sum" and "total number of odd parts" follows by conjugation. (End)

REFERENCES

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

LINKS

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

D. Kim, A. J. Yee, A note on partitions into distinct parts and odd parts, Ramanujan J. 3 (1999), 227-231. [R. J. Mathar, Nov 11 2008]

Wolfdieter Lang, First 11 rows.

FORMULA

a(n, k) = number of partitions of n>=0, which have exactly k odd parts (and possibly even parts) for k from {0, ..., n}.

Sum_{k=0..n} k*T(n,k) = A066897(n). - Emeric Deutsch, Feb 17 2006

G.f.: G(t,x) = 1/Product_{j>=1} (1-t*x^(2*j-1))*(1-x^(2*j)). - Emeric Deutsch, Feb 17 2006

G.f. T(2n+k,k) = g.f. d(n,k) = (1/Product_{j=1..k} (1-x^j)) * g.f. p(n). - Gregory L. Simay, Oct 31 2015

T(n,k) = T(n-1,k-1) + T(n-2k,k). - Gregory L. Simay, Nov 01 2015

EXAMPLE

The triangle a(n,k) begins:

n\k 0  1  2  3  4  5  6  7  8  9 10

0:  1

1:  0  1

2:  1  0  1

3:  0  2  0  1

4:  2  0  2  0  1

5:  0  4  0  2  0  1

6:  3  0  5  0  2  0  1

7:  0  7  0  5  0  2  0  1

8:  5  0  9  0  5  0  2  0  1

9:  0 12  0 10  0  5  0  2  0  1

10: 7  0 17  0 10  0  5  0  2  0  1

... Reformatted - Wolfdieter Lang, Apr 28 2013

a(0,0) = 1 because n=0 has no odd part, only one even part, 0, by definition. a(5,3) = 2 because there are two partitions (1,1,3) and (1,1,1,2) of 5 with exactly 3 odd parts.

From Gregory L. Simay, Oct 31 2015: (Start)

T(10,4) = T(2*3+4,4) = d(3,4) = A000712(3) = 10.

T(10,2) = T(2*4+2,2) = d(4,2) = d(4,1)+d(2,1)+d(0,1) = d(4,0)+d(3,0)+d(2,0)+d(1,0)+d(0,0) + d(2,0)+d(1,0)+d(0,0) + d(0,0) = convolution sum p(4)+p(3)+2*p(2)+2*p(1)+3*p(0) = 5+3+2*2+2*1+3*1 = 17.

T(9,1) = T(8,0) + T(7,1) = 5 + 7 = 12.

(End)

MAPLE

g:=1/product((1-t*x^(2*j-1))*(1-x^(2*j)), j=1..20): gser:=simplify(series(g, x=0, 22)): P[0]:=1: for n from 1 to 18 do P[n]:=coeff(gser, x^n) od: for n from 0 to 18 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form # Emeric Deutsch, Feb 17 2006

MATHEMATICA

T[n_, k_] := T[n, k] = Which[n<k, 0, n == k, 1, Mod[n-k+1, 2] == 0, 0, k == 0, Sum[T[Quotient[n, 2], m], {m, 0, n}], True, T[n-1, k-1]+T[n-2*k, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Mar 05 2014, after Paul D. Hanna *)

PROG

(PARI)

{T(n, k)=if(n>=k, if(n==k, 1, if((n-k+1)%2==0, 0, if(k==0, sum(m=0, n, T(n\2, m)), T(n-1, k-1)+T(n-2*k, k)))))}

for(n=0, 20, for(k=0, n, print1(T(n, k), ", ")); print(""))

\\ Paul D. Hanna, Apr 27 2013

CROSSREFS

Row sums gives A000041 (partition numbers). Columns: k=0: A035363 (with zero entries) A000041 (without zero entries), k=1: A000070, k=2: A000097, k=3: A000098, k=4: A000710, 3k>=n: A000712.

Cf. A066897.

Sequence in context: A058685 A029300 A096397 * A263234 A264394 A283310

Adjacent sequences:  A103916 A103917 A103918 * A103920 A103921 A103922

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Mar 24 2005

STATUS

approved

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Last modified June 23 02:54 EDT 2017. Contains 288633 sequences.