A226659
Sum_{k=0..n} A000041( binomial(n,k) ), where A000041(n) is the number of partitions of n.
Original entry on oeis.org
1, 2, 4, 8, 23, 100, 1003, 31382, 5149096, 7091568720, 287786595280763, 539018517346414192796, 1130813038175196801809538188145, 2336855300714703790840987155549462486654700, 7636154577344556445476348286247799105605643795614728449082014
Offset: 0
Equals the row sums of triangle A090011, which begins:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 5, 11, 5, 1;
1, 7, 42, 42, 7, 1;
1, 11, 176, 627, 176, 11, 1;
1, 15, 792, 14883, 14883, 792, 15, 1;
1, 22, 3718, 526823, 4087968, 526823, 3718, 22, 1; ...
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Table[Sum[PartitionsP[Binomial[n,k]],{k,0,n}],{n,0,20}] (* Indranil Ghosh, Feb 21 2017 *)
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{a(n)=sum(k=0,n,numbpart(binomial(n,k)))}
for(n=0,15,print1(a(n),", "))
A129491
Digital sum of the 2^n-th partition number.
Original entry on oeis.org
1, 2, 5, 4, 6, 24, 22, 34, 83, 120, 152, 145, 286, 477, 561, 796, 1271, 1639, 2471, 3598, 5114, 7221, 10283, 14315, 20585, 29110, 40890, 58834, 82319, 115690, 164128, 232044, 328463, 462853, 657811, 927235, 1311605, 1855787, 2629927, 3708205
Offset: 0
a(9) = 120 since P(2^9) = 4453575699570940947378 and 4+4+5+3+5+7+5+6+9+9+5+7+0+9+4+0+9+4+7+3+7+8 = 120.
A079281
Number of compositions of 2^n into distinct parts.
Original entry on oeis.org
1, 1, 3, 19, 435, 74875, 348317763, 294729601581739, 682404222981720262704195, 298417646219775679438413815505895285915, 13661663328896434876017827688479176004409461863714010289523203
Offset: 0
a(2) = 3 since the compositions of 2^2=4 into distinct parts are 4, 3+1 and 1+3.
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b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
`if`(n=m, x^i, `if`(n>m, 0,
expand(b(n, i-1)+`if`(i>n, 0, x*b(n-i, i-1)))))
end:
a:= n->(p->add(coeff(p, x, i)*i!, i=0..degree(p)))(b(2^n$2)):
seq(a(n), n=0..9); # Alois P. Heinz, Apr 27 2014
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b[n_, i_] := b[n, i] = With[{ m = i*(i+1)/2}, If[n==m, x^i, If[n>m, 0, Expand[b[n, i-1] + If[i>n, 0, x*b[n-i, i-1]]]]]]; a[n_] := Function[{p}, Sum[Coefficient[p, x, i]*i!, {i, 0, Exponent[p, x]}]][b[2^n, 2^n]]; Table[a[n], {n, 0, 9}] (* Jean-François Alcover, Oct 05 2015, after Alois P. Heinz *)
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