A035528 -id:A035528 - cp's OEIS frontend

A003602 Kimberling's paraphrases: if n = (2k-1)*2^m then a(n) = k.

1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 18, 5, 19, 10, 20, 3, 21, 11, 22, 6, 23, 12, 24, 2, 25, 13, 26, 7, 27, 14, 28, 4, 29, 15, 30, 8, 31, 16, 32, 1, 33, 17, 34, 9, 35, 18, 36, 5, 37, 19, 38, 10, 39, 20, 40, 3, 41, 21, 42

Offset: 1

N. J. A. Sloane, Mira Bernstein

Fractal sequence obtained from powers of 2.
k occurs at (2*k-1)*A000079(m), m >= 0. - Robert G. Wilson v, May 23 2006
Sequence is T^(oo)(1) where T is acting on a word w = w(1)w(2)..w(m) as follows: T(w) = "1"w(1)"2"w(2)"3"(...)"m"w(m)"m+1". For instance T(ab) = 1a2b3. Thus T(1) = 112, T(T(1)) = 1121324, T(T(T(1))) = 112132415362748. - Benoit Cloitre, Mar 02 2009
Note that iterating the post-numbering operator U(w) = w(1) 1 w(2) 2 w(3) 3... produces the same limit sequence except with an additional "1" prepended, i.e., 1,1,1,2,1,3,2,4,... - Glen Whitney, Aug 30 2023
In the binary expansion of n, first swallow all zeros from the right, then add 1, and swallow the now-appearing 0 bit as well. - Ralf Stephan, Aug 22 2013
Although A264646 and this sequence initially agree in their digit-streams, they differ after 48 digits. - N. J. A. Sloane, Nov 20 2015
"[This is a] fractal because we get the same sequence after we delete from it the first appearance of all positive integers" - see Cobeli and Zaharescu link. - Robert G. Wilson v, Jun 03 2018
From Peter Munn, Jun 16 2022: (Start)
The sequence is the list of positive integers interleaved with the sequence itself. Provided the offset is suitable (which is the case here) a term of such a self-interleaved sequence is determined by the odd part of its index. Putting some of the formulas given here into words, a(n) is the position of the odd part of n in the list of odd numbers.
Applying the interleaving transform again, we get A110963.
(End)
Omitting all 1's leaves A131987 + 1. - David James Sycamore, Jul 26 2022
a(n) is also the smallest positive number not among the terms between a(a(n-1)) and a(n-1) inclusive (with a(0)=1 prepended). - Neal Gersh Tolunsky, Mar 07 2023

			From _Peter Munn_, Jun 14 2022: (Start)
Start of table showing the interleaving with the positive integers:
   n  a(n)  (n+1)/2  a(n/2)
   1    1      1
   2    1               1
   3    2      2
   4    1               1
   5    3      3
   6    2               2
   7    4      4
   8    1               1
   9    5      5
  10    3               3
  11    6      6
  12    2               2
(End)

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Cristian Cobeli and Alexandru Zaharescu, Promenade around Pascal Triangle - Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, 73-98.
J.-P. Delahaye, La marelle arithmétique, Pour la Science, No. 360, October 2007. In French.
Dale Gerdemann, Plotting Adjacent Points in A003602, Kimberling's Paraphrase, YouTube Video, 2015.
Dale Gerdemann, Plotting Adjacent Terms of A003602 Modulo Increasing Powers of 2, YouTube Video, 2015.
Douglas E. Iannucci and Urban Larsson, Game values of arithmetic functions, arXiv:2101.07608 [math.NT], 2021.
Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv preprint arXiv:1608.00862 [math.GM], 2016.
Clark Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.
Clark Kimberling, Fractal sequences
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Matty van-Son, Palindromic sequences of the Markov spectrum, arXiv:1804.10802 [math.NT], 2018.
Eric Weisstein's World of Mathematics, Odd Part
Index entries for sequences related to binary expansion of n

a(n) is the index of the column in A135764 where n appears (see also A054582).
Cf. A000079, A000265, A001511, A003603, A003961, A014577 (with offset 1, reduction mod 2), A025480, A035528, A048673, A101279, A110963, A117303, A126760, A181988, A220466, A249745, A253887, A337821 (2-adic valuation).
Cf. also A349134 (Dirichlet inverse), A349135 (sum with it), A349136 (Möbius transform), A349431, A349371 (inverse Möbius transform).
Cf. A065120, A131987.
Cf. A264646.
Cf. A123390, A118319.

a003602 = (`div` 2) . (+ 1) . a000265
-- Reinhard Zumkeller, Feb 16 2012, Oct 14 2010

import Data.List (transpose)
a003602 = flip div 2 . (+ 1) . a000265
a003602_list = concat $ transpose [[1..], a003602_list]
-- Reinhard Zumkeller, Aug 09 2013, May 23 2013

A003602:=proc(n) options remember: if n mod 2 = 1 then RETURN((n+1)/2) else RETURN(procname(n/2)) fi: end proc:
seq(A003602(n), n=1..83); # Pab Ter
nmax := 83: for m from 0 to ceil(simplify(log[2](nmax))) do for k from 1 to ceil(nmax/(m+2)) do a((2*k-1)*2^m) := k od: od: seq(a(k), k=1..nmax); # Johannes W. Meijer, Feb 04 2013
A003602 := proc(n)
    a := 1;
    for p in ifactors(n)[2] do
        if op(1,p) > 2 then
            a := a*op(1,p)^op(2,p) ;
        end if;
    end do  :
    (a+1)/2 ;
end proc: # R. J. Mathar, May 19 2016

a[n_] := Block[{m = n}, While[ EvenQ@m, m /= 2]; (m + 1)/2]; Array[a, 84] (* or *)
a[1] = 1; a[n_] := a[n] = If[OddQ@n, (n + 1)/2, a[n/2]]; Array[a, 84] (* Robert G. Wilson v, May 23 2006 *)
a[n_] := Ceiling[NestWhile[Floor[#/2] &, n, EvenQ]/2]; Array[a, 84] (* Birkas Gyorgy, Apr 05 2011 *)
a003602 = {1}; max = 7; Do[b = {}; Do[AppendTo[b, {k, a003602[[k]]}], {k, Length[a003602]}]; a003602 = Flatten[b], {n, 2, max}]; a003602 (* L. Edson Jeffery, Nov 21 2015 *)

A003602(n)=(n/2^valuation(n,2)+1)/2; /* Joerg Arndt, Apr 06 2011 */

import math
def a(n): return (n/2**int(math.log(n - (n & n - 1), 2)) + 1)/2 # Indranil Ghosh, Apr 24 2017

def A003602(n): return (n>>(n&-n).bit_length())+1 # Chai Wah Wu, Jul 08 2022

(define (A003602 n) (let loop ((n n)) (if (even? n) (loop (/ n 2)) (/ (+ 1 n) 2)))) ;; Antti Karttunen, Feb 04 2015

a(n) = (A000265(n) + 1)/2.
a((2*k-1)*2^m) = k, for m >= 0 and k >= 1. - Robert G. Wilson v, May 23 2006
Inverse Weigh transform of A035528. - Christian G. Bower
G.f.: 1/x * Sum_{k>=0} x^2^k/(1-2*x^2^(k+1) + x^2^(k+2)). - Ralf Stephan, Jul 24 2003
a(2*n-1) = n and a(2*n) = a(n). - Pab Ter (pabrlos2(AT)yahoo.com), Oct 25 2005
a(A118413(n,k)) = A002024(n,k); = a(A118416(n,k)) = A002260(n,k); a(A014480(n)) = A001511(A014480(n)). - Reinhard Zumkeller, Apr 27 2006
Ordinal transform of A001511. - Franklin T. Adams-Watters, Aug 28 2006
a(n) = A249745(A126760(A003961(n))) = A249745(A253887(A048673(n))). That is, this sequence plays the same role for the numbers in array A135764 as A126760 does for the odd numbers in array A135765. - Antti Karttunen, Feb 04 2015 & Jan 19 2016
G.f. satisfies g(x) = g(x^2) + x/(1-x^2)^2. - Robert Israel, Apr 24 2015
a(n) = A181988(n)/A001511(n). - L. Edson Jeffery, Nov 21 2015
a(n) = A025480(n-1) + 1. - R. J. Mathar, May 19 2016
a(n) = A110963(2n-1) = A349135(4*n). - Antti Karttunen, Apr 18 2022
a(n) = (1 + n)/2, for n odd; a(n) = a(n/2), for n even. - David James Sycamore, Jul 28 2022
a(n) = n/2^A001511(n) + 1/2. - Alan Michael Gómez Calderón, Oct 06 2023
a(n) = A123390(A118319(n)). - Flávio V. Fernandes, Mar 02 2025

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 25 2005

A262811 Expansion of Product_{k>=1} 1/(1-x^(2k-1))^(2k-1).

Original entry on oeis.org

1, 1, 1, 4, 4, 9, 15, 22, 37, 56, 92, 133, 210, 310, 466, 696, 1013, 1495, 2160, 3141, 4495, 6462, 9172, 13024, 18387, 25840, 36213, 50500, 70280, 97302, 134522, 185105, 254245, 347938, 475036, 646676, 878145, 1189468, 1607095, 2166672, 2913794, 3910741

Offset: 0

Vaclav Kotesovec, Oct 03 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A000219, A003293, A035528, A161870, A262736, A292038.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(d::even, 0, d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Oct 05 2015

nmax = 60; CoefficientList[Series[Product[1/(1-x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(-1/12 + 3*Zeta(3)^(1/3)*n^(2/3)/2) * A * Zeta(3)^(5/36) / (2^(2/3) * sqrt(3*Pi) * n^(23/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A050999(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 09 2017

A262876 Expansion of Product_{k>=1} 1/(1-x^(3*k-1))^k.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 2, 4, 2, 7, 6, 7, 12, 12, 16, 26, 22, 35, 44, 47, 68, 84, 88, 133, 146, 176, 238, 267, 324, 431, 468, 604, 746, 842, 1068, 1296, 1470, 1884, 2202, 2579, 3220, 3753, 4418, 5483, 6294, 7541, 9144, 10554, 12644, 15191, 17480, 21057, 24896

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

a(n) is the number of partitions of n into parts 3*k-1 of k kinds (k>=1).

In general, if s>0, t>0, GCD(s,t)=1 and g.f. = Product_{k>=1} 1/(1 - x^(s*k-t))^k then a(n) ~ s^(t^2/(3*s^2) - 7/18) * n^(t^2/(6*s^2) - 25/36) * exp(d(s,t) - Pi^4 * t^2 / (432*s^2 * Zeta(3)) + Pi^2 * t * 2^(2/3) * s^(2/3) * n^(1/3) / (12 * s^2 * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / (2^(2/3)*s^(2/3))) / (2^(t^2/(6*s^2) + 11/36) * sqrt(3*Pi) * Zeta(3)^(t^2/(6*s^2) - 7/36)), where d(s,t) = Integral_{x=0..infinity} 1/x * (exp(-(s-t)*x)/(1 - exp(-s*x))^2 - 1/(s^2*x^2) - t/(s^2*x) + exp(-x)*(1/12 - t^2/(2*s^2))) dx. - Vaclav Kotesovec, Oct 12 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Vaclav Kotesovec, Graph - The asymptotic ratio (70000 terms)

Cf. A000219, A035528, A255610, A262877, A262878, A262879, A262883, A262946, A263136, A263141.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+3, 3, 'r')=2, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 05 2015

nmax=100; CoefficientList[Series[Product[1/(1-x^(3k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax=100; CoefficientList[Series[E^Sum[1/j*x^(2*j)/(1-x^(3j))^2,{j,1,nmax}],{x,0,nmax}],x]

a(n) ~ Zeta(3)^(19/108) * exp(d1 - Pi^4 / (3888*Zeta(3)) + Pi^2 * n^(1/3) / (2^(4/3)*3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(35/108) * 3^(23/27) * sqrt(Pi) * n^(73/108)), where d1 = A263030 = Integral_{x=0..infinity} 1/x*(exp(-2*x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 1/(9*x) + exp(-x)/36) = -0.188708191979528532376410098649207973592114467268429221509... . - Vaclav Kotesovec, Oct 08 2015

A262877 Expansion of Product_{k>=1} 1/(1-x^(3*k-2))^k.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 6, 9, 9, 13, 19, 23, 28, 42, 51, 62, 84, 108, 127, 170, 219, 261, 328, 427, 512, 632, 807, 987, 1190, 1504, 1838, 2214, 2744, 3374, 4036, 4950, 6060, 7260, 8793, 10748, 12853, 15459, 18766, 22473, 26834, 32425, 38768, 46136, 55376, 66168

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

a(n) is the number of partitions of n into parts 3*k-2 of k kinds (k>=1). - Joerg Arndt, Oct 06 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Vaclav Kotesovec, Graph - The asymptotic ratio (70000 terms)

Cf. A000219, A035528, A255610, A262876, A262878, A262879, A262883, A262947.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+3, 3, 'r')=1, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 05 2015

nmax=100; CoefficientList[Series[Product[1/(1-x^(3k-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax=100; CoefficientList[Series[E^Sum[1/j*x^j/(1-x^(3j))^2,{j,1,nmax}],{x,0,nmax}],x]

a(n) ~ Zeta(3)^(13/108) * exp(d2 - Pi^4 / (972*Zeta(3)) + Pi^2 * n^(1/3) / (2^(1/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(41/108) * 3^(20/27) * sqrt(Pi) * n^(67/108)), where d2 = A263031 = Integral_{x=0..infinity} 1/x*(exp(-x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 2/(9*x) - 5*exp(-x)/36) = -0.0145374291832840336050202945022620903605414975934644413815... . - Vaclav Kotesovec, Oct 08 2015

A262878 Expansion of Product_{k>=1} (1+x^(3*k-1))^k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 2, 3, 0, 4, 4, 1, 10, 5, 6, 16, 6, 14, 28, 10, 32, 40, 18, 63, 60, 42, 112, 83, 84, 187, 124, 172, 300, 186, 320, 456, 302, 581, 684, 507, 982, 1004, 874, 1624, 1476, 1508, 2566, 2174, 2582, 3981, 3262, 4338, 6002, 4945, 7138, 8947, 7660

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

In general, if s>0, t>0, GCD(s,t)=1 and g.f. = Product_{k>=1} (1 + x^(s*k-t))^k then a(n) ~ 2^(t^2/(2*s^2) - 3/4) * s^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4 * t^2 / (1296 * s^2 * Zeta(3)) + Pi^2 * t * 2^(1/3) * 3^(2/3) * s^(2/3) * n^(1/3) / (36 * s^2 * Zeta(3)^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (2^(4/3) * s^(2/3)) ) / (3^(1/3) * s * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Oct 12 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A026007, A027346, A035528, A262876, A262877, A262879, A262884, A262948, A263138, A263145.

with(numtheory):
b:= n-> `if`(n<3, n-1, (p-> [0, -r, 2*r, 0, 0, 2*r+1][p]
         )(1+irem(n+3, 6, 'r'))):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 05 2015

nmax=100; CoefficientList[Series[Product[(1+x^(3k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax=100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(2*j)/(1-x^(3j))^2,{j,1,nmax}],{x,0,nmax}],x]

a(n) ~ exp(2^(-4/3) * 3^(2/3) * Zeta(3)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (2^(5/3)*3^(8/3) * Zeta(3)^(1/3)) - Pi^4/(11664*Zeta(3))) * Zeta(3)^(1/6) / (2^(25/36) * 3^(2/3) * sqrt(Pi) * n^(2/3)).

A263140 Expansion of Product_{k>=1} (1 + x^(2*k-1))^k.

Original entry on oeis.org

1, 1, 0, 2, 2, 3, 4, 5, 10, 11, 16, 20, 31, 39, 50, 71, 93, 124, 154, 211, 271, 357, 449, 587, 762, 968, 1233, 1571, 2021, 2535, 3220, 4049, 5145, 6431, 8070, 10105, 12670, 15784, 19619, 24447, 30348, 37635, 46464, 57532, 70945, 87477, 107456, 132192, 162220

Offset: 0

Vaclav Kotesovec, Oct 10 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A035528, A263149, A263150, A263199, A262878, A263138, A263145, A292037.

nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^j/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^j/(1 - x^(2*j))^2).

a(n) ~ exp(-Pi^4 / (5184*Zeta(3)) + Pi^2 * n^(1/3) / (8 * 3^(4/3) * Zeta(3)^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)/4) * Zeta(3)^(1/6) / (2^(23/24) * 3^(1/3)* sqrt(Pi) * n^(2/3)).

A262879 Expansion of Product_{k>=1} (1+x^(3*k-2))^k.

Original entry on oeis.org

1, 1, 0, 0, 2, 2, 0, 3, 4, 1, 4, 10, 6, 5, 16, 14, 9, 28, 32, 17, 40, 63, 41, 63, 112, 83, 94, 187, 171, 156, 301, 319, 260, 467, 580, 465, 713, 981, 818, 1095, 1627, 1452, 1682, 2584, 2510, 2632, 4047, 4266, 4162, 6181, 7054, 6685, 9396, 11423, 10753, 14132

Offset: 0

Vaclav Kotesovec, Oct 04 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A026007, A027346, A035528, A262876, A262877, A262878, A262884, A262949.

nmax=100; CoefficientList[Series[Product[(1+x^(3k-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax=100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^j/(1-x^(3j))^2,{j,1,nmax}],{x,0,nmax}],x]
Clear[a]; a[n_]:=a[n] = If[n==0, 1, Sum[Sum[d*{0, 2*Floor[d/6] + 1, -Floor[d/6] - 1, 0, 2*Floor[d/6] + 2, 0}[[1 + Mod[d, 6]]], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}]

a(n) ~ exp(2^(-4/3) * 3^(2/3) * Zeta(3)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (2^(2/3) * 3^(8/3) * Zeta(3)^(1/3)) - Pi^4/(2916*Zeta(3))) * Zeta(3)^(1/6) / (2^(19/36) * 3^(2/3) * sqrt(Pi) * n^(2/3)).

A263150 Expansion of Product_{k>=1} 1/(1 - x^(2*k+1))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 1, 3, 2, 5, 6, 7, 11, 12, 21, 22, 34, 38, 59, 67, 95, 118, 155, 198, 252, 330, 409, 540, 662, 867, 1067, 1382, 1705, 2187, 2705, 3430, 4267, 5348, 6666, 8303, 10352, 12812, 15964, 19681, 24467, 30091, 37282, 45769, 56539, 69296, 85304

Offset: 0

Vaclav Kotesovec, Oct 10 2015

nonn

The side effect of this calculation is a formula: Integral_{x=0..infinity} exp(-3*x)/(x*(1-exp(-2*x))^2) - 1/(4*x^3) + 1/(4*x^2) - exp(-x)/(24*x) = log(2)/6 + log(A)/2 - 1/24, where A = A074962 is the Glaisher-Kinkelin constant.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A035528, A052847, A263140, A263149, A263199, A263352, A263395, A263396, A263397.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d-1, 2)=0, (d-1)/2, 0),
       d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # after Alois P. Heinz, Oct 17 2015

nmax = 100; CoefficientList[Series[Product[1/(1-x^(2*k+1))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(3*j)/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} 1/j*x^(3*j)/(1 - x^(2*j))^2).

a(n) ~ sqrt(A) * Zeta(3)^(11/72) * exp(-1/24 - Pi^4/(1728*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(8/3)* Zeta(3)^(1/3)) + 3 * (Zeta(3)/2)^(1/3) * n^(2/3)/2) / (2^(35/72) * sqrt(3*Pi) * n^(47/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A263141 Expansion of Product_{k>=1} 1/(1-x^(5*k-1))^k.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 2, 3, 0, 1, 2, 6, 4, 1, 2, 6, 10, 6, 2, 6, 14, 20, 8, 6, 14, 29, 30, 13, 14, 34, 54, 50, 22, 34, 66, 99, 74, 43, 72, 133, 166, 119, 82, 148, 242, 276, 182, 166, 286, 438, 442, 301, 316, 541, 744, 701, 494, 608, 976, 1255

Offset: 0

Vaclav Kotesovec, Oct 10 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A263142, A263143, A263144, A263178, A263145, A035528, A262876, A263136.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+5, 5, 'r')=4, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100); # after Alois P. Heinz

nmax = 100; CoefficientList[Series[Product[1/(1-x^(5k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(4*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} 1/j*x^(4*j)/(1 - x^(5*j))^2).

a(n) ~ Zeta(3)^(169/900) * exp(d51 - Pi^4/(10800*Zeta(3))+ Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (300 * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(281/900) * 5^(169/450) * sqrt(3*Pi) * n^(619/900)), where d51 = A263178 = Integral_{x=0..infinity} exp(-4*x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 1/(25*x^2) + 19/(300*x*exp(x)) = -0.1269958671388232529452705747311358056... .

A263136 Expansion of Product_{k>=1} 1/(1-x^(4*k-1))^k.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 2, 0, 1, 2, 3, 1, 2, 6, 5, 2, 6, 11, 7, 6, 15, 21, 12, 15, 30, 34, 22, 35, 58, 59, 43, 70, 108, 95, 85, 142, 187, 161, 167, 263, 318, 274, 318, 480, 534, 471, 595, 836, 879, 819, 1081, 1433, 1442, 1429, 1915, 2391, 2365, 2483, 3314, 3947

Offset: 0

Vaclav Kotesovec, Oct 10 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A035528, A262876, A263141, A263137, A263176, A263138.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+4, 4, 'r')=3, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100); # after Alois P. Heinz

nmax = 100; CoefficientList[Series[Product[1/(1-x^(4k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(3*j)/(1 - x^(4*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{j>=1} 1/j*x^(3*j)/(1 - x^(4*j))^2).

a(n) ~ Zeta(3)^(53/288) * exp(d41 - Pi^4/(6912*Zeta(3)) + Pi^2 * n^(1/3) / (48*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3)/4) / (sqrt(3*Pi) * 2^(101/96) * n^(197/288)), where d41 = A263176 = Integral_{x=0..infinity} exp(-3*x)/(x*(1 - exp(-4*x))^2) - 1/(16*x^3) - 1/(16*x^2) + 5/(96*x*exp(x)) = -0.158924147180165035059952001737321408554746599955833696821824808027... .

cp's OEIS Frontend

A003602 Kimberling's paraphrases: if n = (2k-1)*2^m then a(n) = k.

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A262811 Expansion of Product_{k>=1} 1/(1-x^(2*k-1))^(2*k-1).

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A262876 Expansion of Product_{k>=1} 1/(1-x^(3*k-1))^k.

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A262877 Expansion of Product_{k>=1} 1/(1-x^(3*k-2))^k.

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A262878 Expansion of Product_{k>=1} (1+x^(3*k-1))^k.

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A263140 Expansion of Product_{k>=1} (1 + x^(2*k-1))^k.

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A262879 Expansion of Product_{k>=1} (1+x^(3*k-2))^k.

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A262811 Expansion of Product_{k>=1} 1/(1-x^(2k-1))^(2k-1).