A113869
Coefficients in asymptotic expansion of probability that a random pair of elements from the alternating group A_k generates all of A_k.
Original entry on oeis.org
1, -1, -1, -4, -23, -171, -1542, -16241, -194973, -2622610, -39027573, -636225591, -11272598680, -215668335091, -4431191311809, -97316894892644, -2275184746472827, -56421527472282127, -1479397224086870294, -40897073524132164189, -1188896226524012279617
Offset: 0
- Vaclav Kotesovec, Table of n, a(n) for n = 0..420
- L. Babai, The probability of generating the symmetric group, J. Combin. Theory, A52 (1989), 148-153.
- J. Bovey and A. Williamson, The probability of generating the symmetric group, Bull. London Math. Soc. 10 (1978) 91-96.
- J. D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969) 199-205.
- J. D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, vol 11(2), R56.
- Thibault Godin, An analogue to Dixon's theorem for automaton groups, arXiv preprint arXiv:1610.03301 [math.GR], 2016.
- Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.
-
A003319[n_] := A003319[n] = n! - Sum[ k!*A003319[n-k], {k, 1, n-1}]; a[n_] := -Sum[ A003319[i]*StirlingS2[n-1, i-1], {i, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 11 2012, after N. J. A. Sloane *)
A261052
Expansion of Product_{k>=1} (1+x^k)^(k!).
Original entry on oeis.org
1, 1, 2, 8, 31, 157, 915, 6213, 48240, 423398, 4147775, 44882107, 531564195, 6837784087, 94909482330, 1413561537884, 22482554909451, 380269771734265, 6815003300096013, 128992737080703803, 2571218642722865352, 53835084737513866662, 1181222084520177393143
Offset: 0
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(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=25; CoefficientList[Series[Product[(1+x^k)^(k!),{k,1,nmax}],{x,0,nmax}],x]
-
seq(n)={Vec(exp(x*Ser(dirmul(vector(n, n, n!), -vector(n, n, (-1)^n/n)))))} \\ Andrew Howroyd, Jun 22 2018
A305846
Inverse Weigh transform of the Bell numbers (A000110).
Original entry on oeis.org
1, 2, 3, 11, 34, 138, 610, 2976, 15612, 87905, 526274, 3334988, 22270254, 156173299, 1146640394, 8791427525, 70227355786, 583283756678, 5027823752930, 44903579714037, 414877600876638, 3959945233249877, 38996757506464858, 395749369601741015, 4134132167178705732
Offset: 1
-
g:= proc(n) option remember; `if`(n=0, 1,
add(binomial(n-1, j-1)*g(n-j), j=1..n))
end:
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; g(n)-b(n, n-1) end:
seq(a(n), n=1..30);
-
g[n_] := g[n] = If[n == 0, 1,
Sum[Binomial[n-1, j-1]*g[n-j], {j, 1, n}]];
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] = g[n] - b[n, n - 1];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 10 2022, after Alois P. Heinz *)
A305870
Product_{n>=1} (1 + x^n)^a(n) = g.f. of A001147 (double factorial of odd numbers).
Original entry on oeis.org
1, 3, 12, 90, 816, 9206, 122028, 1859550, 32002076, 613891800, 12989299596, 300556868286, 7550646317520, 204687481411974, 5955892982437120, 185158929517924710, 6125200081143892800, 214837724609534836158, 7963817560398871790604, 311101285877490394183800, 12773912991134665452205048
Offset: 1
(1 + x) * (1 + x^2)^3 * (1 + x^3)^12 * (1 + x^4)^90 * (1 + x^5)^816 * ... * (1 + x^n)^a(n) * ... = 1 + 1*x + 1*3*x^2 + 1*3*5*x^3 + 1*3*5*7*x^4 + ... + A001147(k)*x^k + ...
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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; doublefactorial(2*n-1)-b(n, n-1) end:
seq(a(n), n=1..23); # Alois P. Heinz, Jun 13 2018
-
nn = 21; f[x_] := Product[(1 + x^n)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1/(1 + ContinuedFractionK[-k x, 1, {k, 1, nn}]), {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
A306153
Inverse Weigh transform of (-1)^n * n!.
Original entry on oeis.org
-1, 1, -4, 18, -92, 572, -4156, 34177, -314368, 3199844, -35703996, 433422067, -5687955724, 80256874912, -1211781887796, 19496946568897, -333041104402860, 6019770246910128, -114794574818830716, 2303332661419442477, -48509766592884311132, 1069983257387132347080
Offset: 1
(1+x)^(-1)*(1+x^2)*(1+x^3)^(-4)*(1+x^4)^18* ... = 1 - x + 2*x^2 - 6*x^3 + 24*x^4 - ... .
-
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:= n-> (-1)^n*n!-b(n, n-1):
seq(a(n), n=1..24); # Alois P. Heinz, Jun 23 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_] := (-1)^n n! - b[n, n - 1] // FullSimplify;
Array[a, 24] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)
A380614
Product_{n>=1} (1 + x^n)^a(n) = Sum_{n>=0} prime(n)# * x^n.
Original entry on oeis.org
2, 5, 20, 155, 1860, 24970, 444060, 8583935, 202071920, 5992773714, 186947632200, 7001535728810, 288868991951760, 12455290280871150, 587972068547997856, 31327583556949986095, 1856116108295418943020, 113366872636395467452840, 7619343577986975410930880, 541957669076266404650853414
Offset: 1
- 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(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= proc(n) option remember; p(n)-b(n, n-1) end:
seq(a(n), n=1..20); # Alois P. Heinz, Jan 28 2025
-
primorial[n_] := Product[Prime[j], {j, 1, n}]; 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] = primorial[n] - b[n, n - 1]; Array[a, 20]
Showing 1-6 of 6 results.
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