A168246
Inverse Weigh transform of n!.
Original entry on oeis.org
1, 2, 4, 19, 92, 576, 4156, 34178, 314368, 3199936, 35703996, 433422071, 5687955724, 80256879068, 1211781887796, 19496946568898, 333041104402860, 6019770247224496, 114794574818830716, 2303332661419442569, 48509766592884311132, 1069983257387168051076
Offset: 1
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= proc(n) option remember; n! -b(n, n-1) end:
seq(a(n), n=1..30); # Alois P. Heinz, Jun 11 2018
-
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := a[n] = n! - b[n, n - 1];
Array[a, 30] (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz *)
-
seq(n)={dirdiv(Vec(log(1+x*Ser(vector(n, n, n!)))), -vector(n, n, (-1)^n/n))} \\ Andrew Howroyd, Jun 22 2018
A261053
Expansion of Product_{k>=1} (1+x^k)^(k^k).
Original entry on oeis.org
1, 1, 4, 31, 289, 3495, 51268, 891152, 17926913, 409907600, 10499834497, 297793199060, 9262502810645, 313457634240463, 11464902463397642, 450646709610954343, 18943070964019019671, 847932498252050293971, 40266255926484893366914, 2021845081107882645459639
Offset: 0
-
m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1+x^k)^(k^k): k in [1..(m+2)]]))); // G. C. Greubel, Nov 08 2018
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(i^i, j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..25); # Alois P. Heinz, Aug 08 2015
-
nmax=20; CoefficientList[Series[Product[(1+x^k)^(k^k),{k,1,nmax}],{x,0,nmax}],x]
-
m=20; x='x+O('x^m); Vec(prod(k=1,m, (1+x^k)^(k^k))) \\ G. C. Greubel, Nov 08 2018
A305869
Expansion of Product_{k>=1} (1 + x^k)^(2*k-1)!!.
Original entry on oeis.org
1, 1, 3, 18, 123, 1098, 11806, 150406, 2218065, 37206485, 699604235, 14572941915, 333037896380, 8283300923765, 222714069807495, 6436292165450693, 198941178161054798, 6548632634238445779, 228705772883364303114, 8446082393596031365629, 328846269698068735291665, 13462627492562640070346824
Offset: 0
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
binomial(doublefactorial(2*i-1), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..23); # Alois P. Heinz, Jun 13 2018
-
nmax = 21; CoefficientList[Series[Product[(1 + x^k)^(2 k - 1)!!, {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (2 d - 1)!!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]
A321522
Expansion of Product_{k>=1} (1 + x^k)^((k-1)!).
Original entry on oeis.org
1, 1, 1, 3, 8, 32, 153, 883, 5980, 46660, 411861, 4057263, 44104688, 524243696, 6762188285, 94055795999, 1403061499362, 22342571084082, 378257158227079, 6783952072695685, 128481050502464062, 2562250926987454694, 53668572808754641369, 1177957644341460946099
Offset: 0
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*binomial((i-1)!, j), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..23); # Alois P. Heinz, Aug 10 2021
-
nmax = 23; CoefficientList[Series[Product[(1 + x^k)^((k - 1)!), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d!, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 23}]
A380613
Expansion of Product_{k>=1} (1 + x^k)^prime(k)#.
Original entry on oeis.org
1, 2, 7, 42, 291, 2970, 36950, 597100, 11070875, 248103940, 7018494836, 215718595582, 7881561212732, 320881902092122, 13754717161317416, 643588827524430916, 33926485821837232397, 1992916854095359256932, 121393059052727838936847, 8107963745977267426512386, 574571379331620422000295082
Offset: 0
- Eric Weisstein's World of Mathematics, Primorial.
-
p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(p(i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..20); # Alois P. Heinz, Jan 28 2025
-
nmax = 20; CoefficientList[Series[Product[(1 + x^k)^Product[Prime[j], {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x]
primorial[n_] := Product[Prime[j], {j, 1, n}]; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(j/d + 1) d primorial[d], {d, Divisors[j]}] a[n - j], {j, 1, n}]/n]; Table[a[n], {n, 0, 20}]
Showing 1-5 of 5 results.
Comments