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 A097304 Triangle of numbers of partitions of n with m parts which are all odd. 3
 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 1, 0, 3, 0, 2, 0, 1, 0, 1, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 1, 0, 4, 0, 3, 0, 2, 0, 1, 0, 1, 0, 3, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 1, 0, 5, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1, 0, 4, 0, 6, 0, 5, 0, 3, 0, 2, 0, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,17 LINKS Álvar Ibeas, First 100 rows, flattened W. Lang, First 10 rows. FORMULA T(n, m) := 0 if 1 <= n < m, else T(n, m) = number of partitions of n with m parts which are all odd. Hence T(2*k, 2*j-1) = 0, k >= 1, k >= j >= 1; T(2*k-1, 2*j) = 0, k >= 1, k-1 >= j >= 1. G.f.: 1/Product_{j>=1} (1 - t*x^(2*j-1)). - Emeric Deutsch, Feb 24 2006 T(n, k) = T(n-1, k-1) + T(n-2*k, k). If n+k is even, T(n, k) = A008284((n+k)/2, k) = A072233((n-k)/2, k); 0 otherwise. - Álvar Ibeas, Jul 25 2020 EXAMPLE [1]; [0,1]; [1,0,1]; [0,1,0,1]; [1,0,1,0,1]; [0,2,0,1,0,1]; ... T(6,2)=2 because 6 = 1+5 = 3+3; T(6,1) = 0 = T(6,3): there are no partitions of 6 into either one or three parts with only odd numbers; T(6,4)=1 from 6 = 1+1+1+3; T(6,6)=1 from 6 = 1+1+1+1+1+1. MAPLE g:=1/product(1-t*x^(2*j-1), j=1..30)-1: gser:=simplify(series(g, x=0, 17)): for n from 1 to 15 do P[n]:=sort(coeff(gser, x^n)) od: seq(seq(coeff(P[n], t^j), j=1..n), n=1..15); # Emeric Deutsch, Feb 24 2006 CROSSREFS Row sums: A000009 (number of partitions of n into odd parts). Cf. A008284 (partitions of n into k parts). Sequence in context: A287179 A236511 A235924 * A136745 A214157 A246720 Adjacent sequences:  A097301 A097302 A097303 * A097305 A097306 A097307 KEYWORD nonn,tabl,easy,changed AUTHOR Wolfdieter Lang, Aug 13 2004 STATUS approved

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Last modified August 10 04:38 EDT 2020. Contains 336368 sequences. (Running on oeis4.)