A258479
Number of partitions of n into two sorts of parts having exactly 9 parts of the second sort.
Original entry on oeis.org
1, 11, 67, 298, 1080, 3379, 9453, 24204, 57658, 129335, 275693, 562454, 1104484, 2097247, 3865383, 6937016, 12154390, 20838939, 35029203, 57829458, 93897437, 150150058, 236723504, 368350864, 566187387, 860416074, 1293614426, 1925547270, 2839214222, 4149449828
Offset: 9
-
b:= proc(n, i) option remember; series(`if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1)*add(x^t*
binomial(j, t), t=0..min(9, j)), j=0..n/i))), x, 10)
end:
a:= n-> coeff(b(n$2), x, 9):
seq(a(n), n=9..40);
A258480
Number of partitions of n into two sorts of parts having exactly 10 parts of the second sort.
Original entry on oeis.org
1, 12, 79, 377, 1457, 4836, 14289, 38493, 96151, 225486, 501180, 1063635, 2168132, 4265393, 8130869, 15067991, 27222865, 48062380, 83093629, 140925603, 234830485, 384989926, 621737584, 990119455, 1556378360, 2416887471, 3710698393, 5636503638, 8476224739
Offset: 10
-
b:= proc(n, i) option remember; series(`if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1)*add(x^t*
binomial(j, t), t=0..min(10, j)), j=0..n/i))), x, 11)
end:
a:= n-> coeff(b(n$2), x, 10):
seq(a(n), n=10..40);
A278464
Total number of parts of the second sort in all partitions of n into two sorts of parts.
Original entry on oeis.org
0, 1, 5, 17, 53, 145, 385, 957, 2333, 5493, 12741, 28941, 65049, 144225, 317229, 691457, 1497901, 3224145, 6906969, 14726701, 31282421, 66211253, 139720445, 294007373, 617154865, 1292516577, 2701451621, 5635565761, 11736442005, 24403092657, 50666528209
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..3309
- William Dugan, Sam Glennon, Paul E. Gunnells, Einar Steingrimsson, Tiered trees, weights, and q-Eulerian numbers, arXiv:1702.02446 [math.CO], Feb 2017
-
b:= proc(n, i) option remember; `if`(n=0, [1/2, 0], `if`(i<1, 0,
b(n, i-1) +`if`(i>n, 0, (p-> p+[0, p[1]])(2*b(n-i, i)))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..35);
-
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*Sum[x^t*Binomial[j, t], {t, 0, j}], {j, 0, n/i}]]]];
a[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]] . Range[0, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 10 2017, after Alois P. Heinz *)
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