A369079
Number of partitions of [n] such that the element sum of each block is odd.
Original entry on oeis.org
1, 1, 1, 2, 4, 10, 28, 96, 320, 1436, 5556, 28768, 129600, 730864, 3756936, 23286784, 132872192, 910013776, 5679982288, 42235062784, 286769980416, 2281079563104, 16732506817280, 141975748567040, 1115928688967680, 10077454948692288, 84383735744758464
Offset: 0
a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 12|3, 1|23.
a(4) = 4: 124|3, 12|34, 14|23, 1|234.
a(5) = 10: 12345, 124|3|5, 12|34|5, 12|3|45, 14|23|5, 1|234|5, 1|23|45, 14|25|3, 1|245|3, 1|25|34.
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b:= proc(n, x, y) option remember; `if`(n=0, `if`(y=0, 1, 0),
`if`(n::odd, b(n-1, x+1, y)+`if`(x>0, x*b(n-1, x-1, y+1), 0)+
`if`(y>0, y*b(n-1, x+1, y-1), 0), b(n-1, x, y+1)+(x+y)*b(n-1, x, y)))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..26);
# second Maple program:
b:= proc(x, y) option remember; `if`(x+y=0, 1,
add(`if`(j::odd, binomial(x-1, j-1)*add(
b(x-j, y-i)*binomial(y, i), i=0..y), 0), j=1..x))
end:
a:= n-> (h-> b(n-h, h))(iquo(n, 2)):
seq(a(n), n=0..26);
A307375
Expansion of Sum_{j>=0} j!*x^j / Product_{k=1..j} (1 - k^2*x).
Original entry on oeis.org
1, 1, 3, 17, 151, 1893, 31499, 666169, 17351967, 543441005, 20079329875, 861908850561, 42439075349543, 2371469004695797, 149022897087857691, 10448429535366899273, 811758520658841809839, 69463012765807086749949, 6511800419610377560644707, 665560984365147223546851985
Offset: 0
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b:= proc(n, x, y) option remember; `if`(n=0, 1, `if`(n::odd, 0,
b(n-1, y, x+1))+b(n-1, y, x)*x+b(n-1, y, x)*y)
end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..19); # Alois P. Heinz, Jun 10 2023
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nmax = 19; CoefficientList[Series[Sum[j! x^j/Product[(1 - k^2 x), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]
A363589
Number of partitions of [2n+1] such that the largest element of each block is odd.
Original entry on oeis.org
1, 2, 8, 56, 584, 8360, 155720, 3633704, 103284296, 3499082408, 138860069192, 6364334129192, 332934707138888, 19681714722718376, 1303617735072968264, 96028608749005335080, 7816178772774327523400, 698943538498179895072424, 68316963055524325115842376
Offset: 0
a(0) = 1: 1.
a(1) = 2: 123, 1|23.
a(2) = 8: 12345, 123|45, 1245|3, 13|245, 145|23, 1|2345, 1|23|45, 1|245|3.
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b:= proc(n, x, y) option remember; `if`(n=0, 1, `if`(n::even, 0,
b(n-1, y, x+1))+b(n-1, y, x)*x+b(n-1, y, x)*y)
end:
a:= n-> b(2*n+1, 0$2):
seq(a(n), n=0..20);
A290384
Number of ordered set partitions of [n] such that the smallest element of each block is odd.
Original entry on oeis.org
1, 1, 1, 3, 5, 23, 57, 355, 1165, 9135, 37313, 352667, 1723605, 19063207, 108468169, 1374019539, 8920711325, 127336119839, 928899673425, 14751357906571, 119445766884325, 2088674728868631, 18588486479073881, 354892573941671363, 3443175067395538605
Offset: 0
a(3) = 3: 123, 12|3, 3|12.
a(4) = 5: 1234, 124|3, 3|124, 12|34, 34|12.
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b:= proc(n, m, t) option remember; `if`(n=0, m!,
add(b(n-1, max(m, j), 1-t), j=1..m+1-t))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);
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b[n_, m_, t_]:=b[n, m, t]=If[n==0, m!, Sum[b[n - 1, Max[m, j], 1 - t], {j, m + 1 - t}]]; Table[b[n, 0, 0], {n, 0, 50}] (* Indranil Ghosh, Jul 30 2017 *)
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{ A290384(n) = (n==0) + sum(m=0,n, sum(k=1,m+1, stirling(m,k-1,2)*(k-1)! * stirling(n-m,k,2)*k! * (-1)^(m+k+1))); } \\ Max Alekseyev, Sep 28 2021
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{ A290384(n) = polcoef(1 + sum(k=1,n, (-1)^(k-1) / binomial(-1/x-1,k-1) / binomial(1/x-1,k) + O(x^(n+1)) ), n); } \\ Max Alekseyev, Sep 23 2021
A363587
Number of partitions of [n] such that in the set of smallest block elements there is an equal number of odd and even terms.
Original entry on oeis.org
1, 0, 1, 2, 6, 16, 63, 246, 1201, 5632, 30776, 166800, 1032537, 6404960, 44200745, 305485130, 2305218366, 17475547664, 143075155975, 1179769331662, 10409877747841, 92570178170528, 873953428860952, 8318955989166944, 83562716138732321, 846729015766650672
Offset: 0
a(0) = 1: () the empty partition.
a(1) = 0.
a(2) = 1: 1|2.
a(3) = 2: 13|2, 1|23.
a(4) = 6: 123|4, 134|2, 13|24, 14|23, 1|234, 1|2|3|4.
a(5) = 16: 1235|4, 123|45, 1345|2, 134|25, 135|24, 13|245, 13|2|4|5, 145|23, 14|235, 15|234, 1|2345, 1|23|4|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
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b:= proc(n, x, y) option remember; `if`(abs(x-y)>2*n, 0,
`if`(n=0, 1, `if`(y=0, 0, b(n-1, y-1, x+1)*y)+
b(n-1, y, x)*x + b(n-1, y, x+1)))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);
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b[n_, x_, y_] := b[n, x, y] = If[Abs[x - y] > 2n, 0, If[n == 0, 1, If[y == 0, 0, b[n-1, y-1, x+1]*y] + b[n-1, y, x]*x + b[n-1, y, x+1]]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 18 2023, after Alois P. Heinz *)
Showing 1-5 of 5 results.
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