A270915 -id:A270915 - cp's OEIS frontend

A008485 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^n.

1, 1, 5, 22, 105, 506, 2492, 12405, 62337, 315445, 1605340, 8207563, 42124380, 216903064, 1119974875, 5796944357, 30068145905, 156250892610, 813310723925, 4239676354650, 22130265931900, 115654632452535, 605081974091875, 3168828466966388, 16610409114771900

Offset: 0

T. Forbes (anthony.d.forbes(AT)googlemail.com)

nonn

Number of partitions of n into parts of n kinds. - Vladeta Jovovic, Sep 08 2002
Main diagonal of A144064. - Omar E. Pol, Jun 27 2012
From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the supercongruence a(p) == p + 1 (mod p^2) holds for all primes p >= 3. Cf. A270913. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A000712, A000716, A023003-A023021, A005758, A006922.

Cf. A109085, A192435, A252782, A255526, A255672, A270913, A270915, A270919.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:= n-> etr(j->n)(n): seq(a(n), n=0..30); # Alois P. Heinz, Sep 09 2008

a[n_] := SeriesCoefficient[ Product[1/(1-x^k)^n, {k, 1, n}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 1, 24}] (* Jean-François Alcover, Feb 24 2015 *)
Table[SeriesCoefficient[1/QPochhammer[x, x]^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 25 2016 *)
Table[SeriesCoefficient[Exp[n*Sum[x^j/(j*(1-x^j)), {j, 1, n}]], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 19 2018 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k +x*O(x^n))^n),n)}

{a(n)=n*polcoeff(log(1/x*serreverse(x*eta(x+x*O(x^n)))), n)} /* Paul D. Hanna, Apr 05 2012 */

a(n) = Sum_{pi} Product_{i=1..n} binomial(k_i+n-1, k_i) where pi runs through all nonnegative solutions of k_1+2*k_2+...+n*k_n=n. a(n) = b(n, n) where b(n, m)= m/n*Sum_{i=1..n} sigma(i)*b(n-i, m) is recurrence for number of partitions of n into parts of m kinds. - Vladeta Jovovic, Sep 08 2002
Equals the logarithmic derivative of A109085, the g.f. of which is (1/x)*Series_Reversion(x*eta(x)). - Paul D. Hanna, Apr 05 2012
Let G(x) = exp( Sum_{n>=1} a(n)*x^n/n ), then G(x) = 1/Product_{n>=1} (1-x^n*G(x)^n) is the g.f. of A109085. - Paul D. Hanna, Apr 05 2012
a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.352701333486642687772415814165..., c = A327279 = 0.26801521271073331568695383828... . - Vaclav Kotesovec, Sep 10 2014

a(0)=1 prepended by Alois P. Heinz, Mar 30 2015

A109085 G.f. A(x) satisfies: A(x) = P(x*A(x)) where P(x) = A(x/P(x)) is the g.f. of the partition numbers A000041.

Original entry on oeis.org

1, 1, 3, 10, 38, 153, 646, 2816, 12585, 57343, 265401, 1244256, 5896512, 28200365, 135935424, 659754072, 3221354296, 15812501100, 77985955410, 386254209762, 1920391362054, 9580985321554, 47951223856445, 240680464689600

Offset: 0

Paul D. Hanna, Jun 18 2005

nonn

a(n) = Sum[Product(1 + n/h(v)^2)]/(n+1), where the product is over all boxes v in the Ferrers diagram of a partition L of n, h(v) is the hook length of v and the summation is over all partitions L of n. Example: a(3)=10 because for the partitions L=(3), (2,1), (1,1,1) of n=3 the hook length multi-sets are {3,2,1}, {3,1,1},{3,2,1}, respectively, the products are (1+3/9)(1+3/4)(1+3/1)=28/3, (1+3/9)(1+3/1)(1+3/1)=64/3, (1+3/9)(1+3/4)(1+3/1)=28/3 and now a(3)=(1/4)(28+64+28)/3=10. - Emeric Deutsch, May 15 2008

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 153*x^5 + 646*x^6 + ...
G.f. satisfies: P(x*A(x)) = A(x) where P(x) is the partition function:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
The g.f. A = A(x) also satisfies the identities:
(1) A(x) = 1/((1-x*A) * (1-x^2*A^2) * (1-x^3*A^3) * (1-x^4*A^4) * ...).
(2) A(x) = 1 + x*A/(1-x*A) + x^2*A^2/((1-x*A)*(1-x^2*A^2)) + x^3*A^3/((1-x*A)*(1-x^2*A^2)*(1-x^3*A^3)) + ...
(3) A(x) = 1 + x*A/(1-x*A)^2 + x^4*A^4/((1-x*A)*(1-x^2*A^2))^2 + x^9*A^9/((1-x*A)*(1-x^2*A^2)*(1-x^3*A^3))^2 + ...
The logarithm of the g.f. is given by:
log(A(x)) = x*A(x)/(1-x*A(x)) + x^2*A(x)^2/(2*(1-x^2*A(x)^2)) + x^3*A(x)^3/(3*(1-x^3*A(x)^3)) + x^4*A(x)^4/(4*(1-x^4*A(x)^4)) + x^5*A(x)^5/(5*(1-x^5*A(x)^5)) + ...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 105*x^4/4 + 506*x^5/5 + 2492*x^6/6 + 12405*x^7/7 + 62337*x^8/8 + ... + A008485(n)*x^n/n + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO]; see p.5 and p.32
Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications,arXiv:0805.1398v1 [math.CO], see p.5
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Cf. A109084, A184362, A008485, A000041; related: A106336.

A109085list[n_] := Module[{m = 1, A = 1 + x}, For[i = 1, i <= n, i++, A = 1/Product[(1 - x^k*(A + x*O[x]^n)^k), {k, 1, n}]]; CoefficientList[A, x][[1 ;; n]]]; A109085list[24] (* Jean-François Alcover, Apr 21 2016, adapted from PARI *)
InverseSeries[x QPochhammer[x] + O[x]^25][[3]] (* Vladimir Reshetnikov, Nov 17 2016 *)
Table[SeriesCoefficient[1/QPochhammer[x, x]^(n+1), {x, 0, n}]/(n+1), {n, 0, 24}] (* Vladimir Reshetnikov, Nov 20 2016 *)

{a(n)=polcoeff(1/x*serreverse(x*eta(x+x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k*(A+x*O(x^n))^k))); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0,n,x^m*A^m/prod(k=1, m, (1-x^k*(A+x*O(x^n))^k)))); polcoeff(A, n)} \\ Paul D. Hanna, Nov 24 2012
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0,sqrtint(n+1),(x*A)^(m^2)/prod(k=1, m, (1-x^k*(A+x*O(x^n))^k)^2))); polcoeff(A, n)} \\ Paul D. Hanna, Nov 24 2012
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,(x^m*A^m/m)/(1-x^m*A^m+x*O(x^n)) )));polcoeff(A,n)} \\ Paul D. Hanna, Jun 01 2011

{A008485(n)=polcoeff(prod(k=1,n,1/(1-x^k +x*O(x^n))^n),n)}
{a(n)=polcoeff(exp(sum(m=1,n,A008485(m)*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, Feb 06 2012

G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion(x*eta(x)), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.
(2) A(x) = 1/G(x) where G(x) is g.f. of A109084.
(3) A(x) = 1 / Product_{n>=1} (1 - x^n*A(x)^n).
(4) A(x) = Sum_{n>=0} x^n*A(x)^n / Product_{k=1..n} (1-x^k*A(x)^k).
(5) A(x) = Sum_{n>=0} (x*A(x))^(n^2) / Product_{k=1..n} (1-x^k*A(x)^k)^2.
(6) A(x) = exp( Sum_{n>=1} (x^n/n) * A(x)^n/(1 - x^n*A(x)^n) ). - Paul D. Hanna, Jun 01 2011
Logarithmic derivative yields A008485, where A008485(n) is the number of partitions of n into parts of n kinds. - Paul D. Hanna, Feb 06 2012
a(n) = ([x^n] 1/((x; x)inf)^(n+1))/(n+1), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(n) ~ c * d^n / n^(3/2), where d = A270915 = 5.3527013334866426877724... and c = A366022 = 0.489635226684303373081541660578468619322416625... . - Vaclav Kotesovec, Nov 21 2016

A270914 Decimal expansion of a constant related to the asymptotics of A270913.

Original entry on oeis.org

4, 5, 0, 2, 4, 7, 6, 7, 4, 7, 6, 1, 7, 3, 5, 4, 4, 8, 7, 7, 3, 8, 5, 9, 3, 9, 3, 2, 7, 0, 0, 7, 8, 4, 4, 0, 6, 7, 6, 3, 1, 2, 8, 7, 5, 6, 0, 9, 1, 6, 2, 1, 6, 3, 3, 4, 6, 4, 5, 4, 0, 4, 2, 4, 0, 8, 8, 8, 4, 0, 3, 2, 7, 9, 0, 6, 7, 7, 3, 2, 0, 2, 2, 1, 9, 2, 0, 6, 9, 6, 2, 5, 2, 5, 5, 1, 1, 4, 5, 3, 7, 2, 9, 6

Offset: 1

Vaclav Kotesovec, Mar 25 2016

This constant is very close to exp(5*Pi/(6*sqrt(2))) / sqrt(2) = 4.502476748630924546525119125234175537729... - Vaclav Kotesovec, May 17 2018

			4.502476747617354487738593932700784406763128756091621633464540424...

Cf. A181315, A206229, A270913, A270915, A303071, A304626, A327280.

RealDigits[1/r /. FindRoot[{2*s == QPochhammer[-1, r*s], r*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/2}, {s, 1/2}, WorkingPrecision -> 120], 10, 105][[1]] (* Vaclav Kotesovec, Sep 26 2023 *)

Equals limit n->infinity A270913(n)^(1/n).

A206228 a(n) = [x^n] Product_{k=1..n} 1/(1 - x^k)^(n-k+1).

Original entry on oeis.org

1, 1, 4, 17, 80, 384, 1887, 9385, 47139, 238488, 1213588, 6204547, 31844710, 163978344, 846741721, 4382945317, 22735196277, 118151632006, 615032941924, 3206257881171, 16736910271178, 87472908459696, 457662760258109, 2396899780970552, 12564645719730297

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Number of partitions of n with 1 kind of n's, 2 kinds of (n-1)'s, ..., n kinds of 1's, see example. [Joerg Arndt, May 17 2013]

			Let [x^n] F(x) denote the coefficient of x^n in F(x); then
a(0) = 1;
a(1) = [x] 1/(1-x) = 1;
a(2) = [x^2] 1/((1-x)^2*(1-x^2)) = 4;
a(3) = [x^3] 1/((1-x)^3*(1-x^2)^2*(1-x^3)) = 17;
a(4) = [x^4] 1/((1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)) = 80; ...
as illustrated below.
The coefficients in Product_{k=1..n} 1/(1-x^k)^(n-k+1) for n=0..9 begin:
n=0: [(1), 0, 0, 0, 0, 0, 0, ...];
n=1: [1,(1), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...];
n=2: [1, 2,(4), 6, 9, 12, 16, 20, 25, 30, 36, 42, ...]; (A002620)
n=3: [1, 3, 8, (17), 33, 58, 97, 153, 233, 342, 489, 681, ...]; (A002625)
n=4: [1, 4, 13, 34, (80), 170, 339, 636, 1141, 1964, 3270, ...];
n=5: [1, 5, 19, 58, 157,(384), 874, 1869, 3803, 7408, 13907, ...];
n=6: [1, 6, 26, 90, 273, 746, (1887), 4474, 10062, 21620, ...];
n=7: [1, 7, 34, 131, 438, 1314, 3632, (9385), 22940, 53466, ...];
n=8: [1, 8, 43, 182, 663, 2158, 6445, 17944, (47139), 117842, ...];
n=9: [1, 9, 53, 244, 960, 3361, 10757, 32008, 89651, (238488), ...]; ...
where the coefficients in parenthesis start this sequence.
Incidentally, the antidiagonal sums in the above table form A206119.
From _Joerg Arndt_, May 17 2013: (Start)
There are a(3)=17 partitions of 3 into 1 kind of 3's, 2 kinds of 2's, and 3 kinds of 1's:
01:  [ 1:0  1:0  1:0  ]
02:  [ 1:0  1:0  1:1  ]
03:  [ 1:0  1:0  1:2  ]
04:  [ 1:0  1:1  1:1  ]
05:  [ 1:0  1:1  1:2  ]
06:  [ 1:0  1:2  1:2  ]
07:  [ 1:0  2:0  ]
08:  [ 1:0  2:1  ]
09:  [ 1:1  1:1  1:1  ]
10:  [ 1:1  1:1  1:2  ]
11:  [ 1:1  1:2  1:2  ]
12:  [ 1:1  2:0  ]
13:  [ 1:1  2:1  ]
14:  [ 1:2  1:2  1:2  ]
15:  [ 1:2  2:0  ]
16:  [ 1:2  2:1  ]
17:  [ 3:0  ]
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A206119, A206229.

Table[SeriesCoefficient[Product[1/(1 - x^k)^(n-k+1), {k, 1, n}], {x, 0, n}], {n, 0, 40}] (* Vaclav Kotesovec, Aug 21 2018 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^n))^(n-k+1)),n)}
for(n=0,41,print1(a(n),", "))

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.3527013334866426877724158141653278798514832712869470973196907560641... and c = 0.2030089852709942695768237484498370155967795685257713505678384193773498... - Vaclav Kotesovec, Aug 21 2018

A303070 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^n.

Original entry on oeis.org

1, 2, 8, 35, 164, 787, 3857, 19147, 96004, 485009, 2465013, 12589315, 64555985, 332158127, 1714001409, 8866730665, 45968787524, 238778897128, 1242417984179, 6474394344503, 33784931507529, 176515163156311, 923265560495737, 4834081924982522, 25334170138318345, 132883719945537587

Offset: 0

Ilya Gutkovskiy, Apr 18 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..300
Index entries for sequences related to partitions

Main diagonal of A210764.

Cf. A000070, A000713, A008485, A144064, A210843.

Table[SeriesCoefficient[1/(1 - x) Product[1/(1 - x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[1/(1 - x) Exp[n Sum[x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 25}]

a(n) = [x^n] (1/(1 - x))*exp(n*Sum_{k>=1} x^k/(k*(1 - x^k))).
a(n) = A210764(n,n) = Sum_{j=0..n} A144064(j,n).
a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.352701333486642687772415814165... and c = 0.4068869940800214657298372785820... - Vaclav Kotesovec, May 19 2018

A109084 G.f. A(x) satisfies: A(x) = 1/G000041(x/A(x)) where G000041(x) is the g.f. of the partition numbers A000041.

Original entry on oeis.org

1, -1, -2, -5, -17, -63, -253, -1062, -4615, -20570, -93538, -432211, -2023567, -9578815, -45767162, -220431025, -1069079067, -5216655257, -25592441875, -126157044454, -624560659184, -3103962569509, -15480272621533, -77450458331100, -388627340240958, -1955249529839424

Offset: 0

Paul D. Hanna, Jun 18 2005

sign

Note: coefficient [x^n] A(x)^n = -A000203(n) (sum of divisors of n) for n>0.

			The initial terms [x^0] through [x^n] of n-th self-convolution
are persistently small:
A^0: 1;
A^1: 1,-1;
A^2: 1,-2,-3;
A^3: 1,-3,-3,-4;
A^4: 1,-4,-2,0,-7;
A^5: 1,-5,0,5,0,-6;
A^6: 1,-6,3,10,3,6,-12;
A^7: 1,-7,7,14,0,7,0,-8;
A^8: 1,-8,12,16,-10,0,-8,8,-15;
A^9: 1,-9,18,15,-27,-9,-21,0,0,-13;

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A109085, A000041, A000203.

(* Calculation of constant c: *) val = Sqrt[r*s^5*(-1 + s/r)*(Log[r/s]^2 / (2*Pi*(2*s^3*(-s*Log[1 - r/s] + ArcTanh[1 - 2*r/s] * (2*r - (r - s)*(Log[1 - r/s] - 2*Log[r/s]))) + (r - s)*(s^3*(2 - 2*Log[1 - r/s] + 3*Log[r/s]) * QPolyGamma[0, 1, r/s] - s^3*QPolyGamma[0, 1, r/s]^2 + s^3*QPolyGamma[1, 1, r/s] + r*Log[r/s]*(r*Log[r/s] * Derivative[0, 2][QPochhammer][r/s, r/s] - 2*s^2*Derivative[0, 0, 1][QPolyGamma][0, 1, r/s])))))] /. FindRoot[{QPochhammer[r/s] == s, (Log[1 - r/s] + QPolyGamma[0, 1, r/s])/Log[r/s] == 1 + (r*Derivative[0, 1][QPochhammer][r/s, r/s])/s^2}, {r, 1/5}, {s, 1/2}, WorkingPrecision -> 1000]; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3] (* Vaclav Kotesovec, Oct 02 2023 *)

a(n)=polcoeff(x/serreverse(x*eta(x+x*O(x^n))),n)

G.f.: A(x) = x/series_reversion(x*eta(x)). G.f.: A(x) = 1/G109085(x) where G109085(x) is g.f. of A109085.

a(n) ~ -c * d^n / n^(3/2), where d = A270915 = 5.35270133348664268777241581416... and c = 0.146705445870000769931272287955221766131167... - Vaclav Kotesovec, May 13 2018

A309986 Convolution of A270913 and A008485.

Original entry on oeis.org

1, 2, 9, 43, 206, 999, 4915, 24372, 121698, 611244, 3085612, 15645347, 79639602, 406809249, 2084567381, 10712007629, 55187254157, 284981396231, 1474729519719, 7646180479889, 39713643612380, 206600871071930, 1076372569004514, 5615363541987786, 29331204404385053

Offset: 0

Vaclav Kotesovec, Aug 26 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A008485, A270913, A270919, A327214, A327215.

A270913[n_] := SeriesCoefficient[Product[(1 + x^k)^n, {k, 1, n}], {x, 0, n}];
A008485[n_] := SeriesCoefficient[Product[1/(1 - x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A008485[k]*A270913[n-k], {k, 0, n}], {n, 0, 20}]

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.352701333486642687... and c = 0.446705640528056029457240607298917821281915554...

A327215 Self-convolution of A008485.

Original entry on oeis.org

1, 2, 11, 54, 279, 1442, 7530, 39474, 207693, 1095522, 5790116, 30650038, 162451560, 861920492, 4577055823, 24323292984, 129338944225, 688128700422, 3662798123481, 19504467792378, 103899170100154, 553642311668244, 2951010332435759, 15733439067954134

Offset: 0

Vaclav Kotesovec, Aug 26 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000712, A008485, A304444, A309986, A327214.

A008485[n_]:=SeriesCoefficient[Product[1/(1-x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A008485[k]*A008485[n-k], {k, 0, n}], {n, 0, 25}]

a(n) ~ c^2 * Pi * d^n, where d = A270915 = 5.352701333486642687772415814165... and c = A327279 = 0.26801521271073331568695383828... (see A008485).

A327279 Decimal expansion of a constant related to A008485 and A327215.

Original entry on oeis.org

2, 6, 8, 0, 1, 5, 2, 1, 2, 7, 1, 0, 7, 3, 3, 3, 1, 5, 6, 8, 6, 9, 5, 3, 8, 3, 8, 2, 8, 0, 3, 2, 8, 6, 7, 9, 5, 0, 0, 6, 6, 6, 7, 5, 7, 2, 4, 2, 0, 3, 9, 4, 2, 6, 4, 4, 5, 9, 0, 4, 1, 5, 8, 4, 6, 9, 5, 3, 9, 0, 9, 4, 9, 9, 2, 6, 7, 0, 6, 0, 0, 5, 4, 3, 3, 5, 0, 1, 7, 4, 3, 9, 4, 2, 2, 3, 1, 2, 9, 5, 4, 0, 8, 3, 2, 1

Offset: 0

Vaclav Kotesovec, Aug 28 2019

			0.26801521271073331568695383828032867950066675724203942644590415846953909499267...

Cf. A008485, A270915, A327215, A327280, A366022.

val = Sqrt[(1 - r*s)*(Log[r*s]^2/(2*Pi*(4*ArcTanh[1 - 2*r*s]*(r*s + (-1 + r*s)*Log[r*s]) - 2*(1 + (-1 + r*s)*ArcTanh[1 - 2*r*s])*Log[1 - r*s] + (-1 + r*s)*(2 + 3*Log[r*s] - 2*Log[1 - r*s]) * QPolyGamma[0, 1, r*s] + (1 - r*s)* QPolyGamma[0, 1, r*s]^2 + (-1 + r*s)*(QPolyGamma[1, 1, r*s] + r*s*Log[r*s]*(r*s^2*Log[r*s] * Derivative[0, 2][QPochhammer][r*s, r*s] - 2*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s])))))] /. FindRoot[{QPochhammer[r*s] == 1/s, 1/s + r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == (Log[1 - r*s] + QPolyGamma[0, 1, r*s])/(s*Log[r*s])}, {r, 1/5}, {s, 2}, WorkingPrecision -> 1000]; RealDigits[Chop[val], 10, -Floor[Log[10, Abs[Im[val]]]] - 3][[1]] (* Vaclav Kotesovec, Oct 02 2023 *)

Equals limit_{n->infinity} A008485(n) * sqrt(n) / A270915^n.

A192435 Number of terms in n-th derivative of a function composed with itself n times.

Original entry on oeis.org

1, 2, 6, 26, 110, 532, 2541, 12644, 63024, 318857, 1618947, 8277062, 42453073, 218597485, 1128527057, 5841301830, 30297014746, 157442596130, 819511659381, 4272054888643, 22299423992018, 116539878029773, 609718298887977, 3193136462042241, 16737951567806110

Offset: 1

Alois P. Heinz, Aug 18 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..600
W. C. Yang, Derivatives are essentially integer partitions, Discrete Mathematics, 222(1-3), July 2000, 235-245.

Main diagonal of A022818.

Cf. A008485.

A:= proc(n, k) option remember;
      `if`(k=1, 1, add(b(n, n, i)*A(i, k-1), i=0..n))
    end:
b:= proc(n, i, k) option remember; `if`(n A(n, n):
seq(a(n), n=1..40);

A[n_, k_] := A[n, k] = If[k == 1, 1, Sum[b[n, n, i]*A[i, k-1], {i, 0, n}]]; b[n_, i_, k_] := b[n, i, k] = If[nJean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.35270133348664..., c = 0.0504640078963302151598181537452... . - Vaclav Kotesovec, Sep 03 2014, updated May 19 2018

cp's OEIS Frontend

A008485 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^n.

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A109085 G.f. A(x) satisfies: A(x) = P(x*A(x)) where P(x) = A(x/P(x)) is the g.f. of the partition numbers A000041.

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A270914 Decimal expansion of a constant related to the asymptotics of A270913.

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A206228 a(n) = [x^n] Product_{k=1..n} 1/(1 - x^k)^(n-k+1).

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A303070 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^n.

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A109084 G.f. A(x) satisfies: A(x) = 1/G000041(x/A(x)) where G000041(x) is the g.f. of the partition numbers A000041.

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A309986 Convolution of A270913 and A008485.

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