A005797 -id:A005797 - cp's OEIS frontend

A002057 Fourth convolution of Catalan numbers: a(n) = 4binomial(2n+3,n)/(n+4).

1, 4, 14, 48, 165, 572, 2002, 7072, 25194, 90440, 326876, 1188640, 4345965, 15967980, 58929450, 218349120, 811985790, 3029594040, 11338026180, 42550029600, 160094486370, 603784920024, 2282138106804, 8643460269248, 32798844771700, 124680849918352

Offset: 0

N. J. A. Sloane

a(n) is sum of the (flattened) list obtained by the iteration of: replace each integer k with the list 0,...,k+1 on the starting value 0. Length of this list is Catalan(n) or A000108. - Wouter Meeussen, Nov 11 2001
a(n-2) is the number of n-th generation vertices in the tree of sequences with unit increase labeled by 3 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003
Number of standard tableaux of shape (n+2,n-1). - Emeric Deutsch, May 30 2004
a(n) = CatalanNumber(n+3) - 2*CatalanNumber(n+2). Proof. From its definition as a convolution of Catalan numbers, a(n) counts lists of 4 Dyck paths of total size (semilength) = n. Connect the 4 paths by 3 upsteps (U) and append 3 downsteps (D). This is a reversible procedure. So a(n) is also the number of Dyck (n+3)-paths that end DDD (D for downstep). Let C(n) denote CatalanNumber(n) (A000108). Since C(n+3) is the total number of Dyck (n+3)-paths and C(n+2) is the number that end UD, we have (*) C(n+3) - C(n+2) is the number of Dyck (n+3)-paths that end DD. Also, (**) C(n+2) is the number of Dyck (n+3)-paths that end UDD (change the last D in a Dyck (n+2)-path to UDD). Subtracting (**) from (*) yields a(n) = C(n+3) - 2C(n+2) as claimed. - David Callan, Nov 21 2006
Convolution square of the Catalan sequence without one of the initial "1"'s: (1 + 4x + 14x^2 + 48x^3 + ...) = (1/x^2) * square(x + 2x^2 + 5x^3 + 14x^4 + ...)
a(n) is the number of binary trees with n+3 internal nodes in which both subtrees of the root are nonempty.  Cf. A068875 [Sedgewick and Flajolet]. - Geoffrey Critzer, Jan 05 2013
With offset 4, a(n) is the number of permutations on {1,2,...,n} that are 123-avoiding, i.e., do not contain a three-term monotone subsequence, for which the first ascent is at positions (4,5); for example, there are 48 123-avoiding permutations on n=7 for which the first ascent is at spots (4,5). See Connolly link. There it is shown in general that the k-th Catalan Convolution is the number of 123-avoiding permutations for which the first ascent is at (k, k+1). (For n=k, the first ascent is defined to be at positions (k,k+1) if the permutation is the decreasing permutation with no ascents.) - Anant Godbole, Jan 17 2014
With offset 4, a(n) is the number of permutations on {1,2,...,n} that are 123-avoiding and for which the integer n is in the 4th spot; see Connolly link. - Anant Godbole, Jan 17 2014
a(n) is the number of North-East lattice paths from (0,0) to (n+2,n+2) that have exactly one east step below the subdiagonal y = x-1. Details can be found in Section 3.1 in Pan and Remmel's link. - Ran Pan, Feb 04 2016
a(n) is the number of North-East lattice paths from (0,0) to (n+2,n+2) that bounce off the diagonal y = x to the right exactly once but do not bounce off y = x to the left. Details can be found in Section 4.2 in Pan and Remmel's link. - Ran Pan, Feb 04 2016
a(n) is the number of North-East lattice paths from (0,0) to (n+2,n+2) that horizontally cross the diagonal y = x exactly once but do not cross the diagonal vertically. Details can be found in Section 4.3 in Pan and Remmel's link. - Ran Pan, Feb 04 2016
Apparently also Young tableaux of (non-partition) shape [n+1, 1, 1, n+1], see example file. - Joerg Arndt, Dec 30 2023

			From _Peter Bala_, Apr 14 2017: (Start)
This sequence appears on the main diagonal of a generalized Catalan triangle. Construct a lower triangular array (T(n,k)), n,k >= 0 by placing the sequence [0,0,0,1,1,1,1,...] in the first column and then filling in the remaining entries in the array using the rule T(n,k) = T(n,k-1) + T(n-1,k). The resulting array begins
  n\k| 0 1  2  3  4   5   6   7  ...
  ---+-------------------------------
   0 | 0
   1 | 0 0
   2 | 0 0  0
   3 | 1 1  1  1
   4 | 1 2  3  4  4
   5 | 1 3  6 10 14  14
   6 | 1 4 10 20 34  48  48
   7 | 1 5 15 35 69 117 165 165
   ...
(see Tedford 2011; this is essentially the array C_4(n,k) in the notation of Lee and Oh). Compare with A279004. (End)

Pierre de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 11, coefficients of P_4(z).
C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., Vol. 14 (1922), pp. 55-62, 122-138 and 143-146.
Robert Sedgewick and Phillipe Flajolet, Analysis of Algorithms, Addison Wesley, 1996, page 225.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Fung Lam, Table of n, a(n) for n = 0..1000 (first 100 terms were computed by T. D. Noe)
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
Joerg Arndt, The a(3)=48 Young tableaux for shape [4,1,1,4].
Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Jean-Luc Baril and Helmut Prodinger, Enumeration of partial Lukasiewicz paths, arXiv:2205.01383 [math.CO], 2022.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Julia E. Bergner, Cedric Harper, Ryan Keller, and Mathilde Rosi-Marshall, Action graphs, planar rooted forests, and self-convolutions of the Catalan numbers, arXiv:1807.03005 [math.CO], 2018.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
Gi-Sang Cheon, Hana Kim, and Louis W. Shapiro, Mutation effects in ordered trees, arXiv preprint arXiv:1410.1249 [math.CO], 2014 (see page 4).
Samuel Connolly, Zachary Gabor, and Anant Godbole, The location of the first ascent in a 123-avoiding permutation, arXiv:1401.2691 [math.CO], 2014.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin, and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., Vol. 35, No. 4 (1995) pp. 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin, and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., Vol. 35, No. 4 (1995) pp. 743-751. [Annotated scanned copy]
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 31.
Harold M. Edwards, A Normal Form for Elliptic Curves, Bulletin of the A.M.S., Vol. 44, No. 3 (2007), pp. 393-422.
Richard K. Guy, Letter to N. J. A. Sloane, May 1990
Richard K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seq., Vol. 3 (2000), Article 00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., Vol. 14 (1922), pp. 55-62, 122-138 and 143-146. [Annotated scanned copy]
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart., Vol. 38, No. 5 (2000), pp. 408-419. See Eq.(3).
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Nik Lygeros and Olivier Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq., Vol. 13 (2010), Article 10.7.4.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
John Riordan, Letter to N. J. A. Sloane, Nov 10 1970.
John Riordan, Letter of 04/11/74.
L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), pp. 83-90.
L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), pp. 83-90. [Annotated scanned copy]
Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., Vol. 10 (2003), Article N5.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.
Steven J. Tedford, Combinatorial interpretations of convolutions of the Catalan numbers, Integers, Vol. 11 (2011), Article A3.
Wen-Jin Woan, Lou Shapiro, and D. G. Rogers, The Catalan numbers, the Lebesgue integral and 4^{n-2}, Amer. Math. Monthly, Vol. 104, No. 10 (1997), pp. 926-931.

T(n, n+4) for n=0, 1, 2, ..., array T as in A047072. Also a diagonal of A059365 and of A009766.
Cf. A001003.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108, A000245, A000344, A003517, A000588, A003518, A003519, A001392, A001622.
Cf. A145596 (row sums), A279004.

List([0..25],n->4*Binomial(2*n+3,n)/(n+4)); # Muniru A Asiru, Mar 05 2018

[4*Binomial(2*n+3,n)/(n+4): n in [0..30]]; // Vincenzo Librandi, Feb 04 2016

a := n -> 32*4^n*GAMMA(5/2+n)*(1+n)/(sqrt(Pi)*GAMMA(5+n)):
seq(a(n),n=0..23); # Peter Luschny, Dec 14 2015
A002057List := proc(m) local A, P, n; A := [1]; P := [1,1,1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A002057List(27); # Peter Luschny, Mar 26 2022

Table[Plus@@Flatten[Nest[ #/.a_Integer:> Range[0, a+1]&, {0}, n]], {n, 0, 10}]
Table[4 Binomial[2n+3,n]/(n+4),{n,0,30}] (* or *) CoefficientList[ Series[ (1-Sqrt[1-4 x]+2 x (-2+Sqrt[1-4 x]+x))/(2 x^4),{x,0,30}],x] (* Harvey P. Dale, May 05 2011 *)

{a(n) = if( n<0, 0, n+=2; 2*binomial(2*n, n-2) / n)}; /* Michael Somos, Jul 31 2005 */

x='x+O('x^100); Vec((1-(1-4*x)^(1/2)+2*x*(-2+(1-4*x)^(1/2)+x))/(2*x^4)) \\ Altug Alkan, Dec 14 2015

[2*(n+1)*catalan_number(n+2)/(n+4) for n in (0..30)] # G. C. Greubel, May 27 2022

a(n) = A033184(n+4, 4) = 4*binomial(2*n+3, n)/(n+4) = 2*(n+1)*A000108(n+2)/(n+4).
G.f.: c(x)^4 with c(x) g.f. of A000108 (Catalan).
Row sums of A145596. Column 4 of A033184. By specializing the identities for the row polynomials given in A145596 we obtain the results a(n) = Sum_{k = 0..n} (-1)^k*binomial(n+1,k+1)*a(k)*4^(n-k) and a(n) = Sum_{k = 0..floor(n/2)} binomial(n+1,2*k+1) * Catalan(k+1) * 2^(n-2*k). From the latter identity we can derive the congruences a(2n+1) == 0 (mod 4) and a(2n) == Catalan(n+1) (mod 4). It follows that a(n) is odd if and only if n = (2^m - 4) for some m >= 2. - Peter Bala, Oct 14 2008
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=3, a(n-3) = (-1)^(n-3) * coeff(charpoly(A,x), x^3). - Milan Janjic, Jul 08 2010
G.f.: (1-sqrt(1-4*x) + 2*x*(-2+sqrt(1-4*x) + x))/(2*x^4). - Harvey P. Dale, May 05 2011
a(n+1) = A214292(2*n+4,n). - Reinhard Zumkeller, Jul 12 2012
D-finite with recurrence: (n+4)a(n) = 8*(2*n-1)*a(n-3) - 20*(n+1)*a(n-2) + 4*(2*n+5)*a(n-1). - Fung Lam, Jan 29 2014
D-finite with recurrence: (n+4)*a(n) - 2*(3*n+7)*a(n-1) + 4*(2*n+1)*a(n-2) = 0. - R. J. Mathar, Jun 03 2014
Asymptotics: a(n) ~ 4^(n+3)/sqrt(4*Pi*n^3). - Fung Lam, Mar 31 2014
a(n) = 32*4^n*Gamma(5/2+n)*(1+n)/(sqrt(Pi)*Gamma(5+n)). - Peter Luschny, Dec 14 2015
a(n) = C(n+1) - 2*C(n) where C is Catalan number A000108. Yuchun Ji, Oct 18 2017 [Note: Offset is off by 2]
E.g.f.: d/dx ( 2*exp(2*x)*BesselI(2,2*x)/x ). - Ilya Gutkovskiy, Nov 01 2017
From Bradley Klee, Mar 05 2018: (Start)
With F(x) = 16/(1+sqrt(1-4*x))^4 g.f. of A002057, xi(x) = F(x/4)*(x/4)^2, K(16*x) = 2F1(1/2,1/2;1;16*x) g.f. of A002894, q(x) g.f. of A005797, and q'(x) g.f. of A274344:
  K(x) = (1+sqrt(xi(x)))*K(xi(x)).
  2*K(1-x) = (1+sqrt(xi(x)))*K(1-xi(x)).
  q(x) = sqrt(q(xi(16*x)/16)) = q'(xi(16*x)/16)/sqrt(xi(16*x)/16). (End)
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=0} 1/a(n) = 5/4 + Pi/(18*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 183*log(phi)/(25*sqrt(5)) - 77/100, where phi is the golden ratio (A001622). (End)
a(n) = Integral_{x=0..4} x^n*W(x) dx where W(x) = -x^(3/2)*(1 - x/2)*sqrt(4 - x)/Pi, defined on the open interval (0,4). - Karol A. Penson, Nov 13 2022

A005798 Expansion of (theta_2(q)/theta_3(q))^4/16 in powers of q.

Original entry on oeis.org

0, 1, -8, 44, -192, 718, -2400, 7352, -20992, 56549, -145008, 356388, -844032, 1934534, -4306368, 9337704, -19771392, 40965362, -83207976, 165944732, -325393024, 628092832, -1194744096, 2241688744, -4152367104, 7599231223, -13749863984

Offset: 0

N. J. A. Sloane

When multiplied by 16, this is the q-expansion of the automorphic function lambda (see A115977) [see Erdelyi].

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q - 8*q^2 + 44*q^3 - 192*q^4 + 718*q^5 - 2400*q^6 + 7352*q^7 - 20992*q^8 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 121.
A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, p. 23, Eq. (37).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
A. Kneser, Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen, J. reine u. angew. Math. 157, 1927, 209 - 218, p.213, second formula.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Elliptic Lambda Function.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

If initial 0 is omitted and sequence begins with a(0) = 1, then this is the convolution of A001938 with itself. G.f.s are related by a(x)=x*A001938(x)^2. Reversion of A005797.

Cf. A007248, A029845, A115977, A128692.

with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: aa:=etr (n-> [ -8,16,-8,0] [modp(n-1,4)+1]): a:= n->aa(n-1): seq (a(n), n=0..26);  # Alois P. Heinz, Sep 08 2008

a[ n_] := SeriesCoefficient[ InverseEllipticNomeQ[ x] / 16, {x, 0, n}]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^8, {q, 0, n}]; (* Michael Somos, May 10 2014 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^4] / QPochhammer[ -q])^8, {q, 0, n}]; (* Michael Somos, May 10 2014 *)
a[ n_] := SeriesCoefficient[ q (Product[ 1 - q^k, {k, 4, n - 1, 4}] / Product[ 1 - (-q)^k, {k, n - 1}])^8, {q, 0, n}]; (* Michael Somos, May 10 2014 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; aa = etr[Function[{n}, {-8, 16, -8, 0}[[Mod[n-1, 4]+1]]]]; a[n_] := aa[n-1]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

{a(n) = my(A, m); if( n<1, 0, m=1; A = x + O(x^2); while( m

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^8, n))}; /* Michael Somos, Jul 16 2005 */

Expansion of elliptic lambda / 16 = m / 16 = (k / 4)^2 in powers of the nome q.
Expansion of q * (psi(q) / phi(q))^8 = q * (psi(q^2) / psi(q))^8 = q * (psi(-q) / phi(-q^2))^8 = q * (chi(-q) / chi(-q^2)^2)^8 = q / (chi(q) * chi(-q^2))^8 = q / (chi(-q) * chi(q)^2)^8 = q * (psi(q^2) / phi(q))^4 = q * (f(-q^4) / f(q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jun 13 2011
Expansion of eta(q)^8 * eta(q^4)^16 / eta(q^2)^24 in powers of q.
Euler transform of period 4 sequence [-8, 16, -8, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v + 16*u*v - 32*u^2*v + 256*(u*v)^2. - Michael Somos, Mar 19 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1 / 16) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128692. - Michael Somos, May 10 2014
G.f.: q * Product( (1 + q^(2*n)) / (1 + q^(2*n - 1)), n=1..inf )^8.
a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n))/(512*n^(3/4)). - Vaclav Kotesovec, Jul 10 2016
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 17/16 - 3*sqrt(2)/4, verified to 27000 digits (10000 terms). - Simon Plouffe, Mar 01 2021

Definition simplified by N. J. A. Sloane, Sep 25 2011

A002639 Numerators of expansion of Jacobi nome q in parameter m.

Original entry on oeis.org

0, 1, 1, 21, 31, 6257, 10293, 279025, 483127, 435506703, 776957575, 22417045555, 40784671953, 9569130097211, 17652604545791, 523910972020563, 976501268709949, 935823746406530603, 1758220447807291611

Offset: 0

N. J. A. Sloane

			q = 1/16*m + 1/32*m^2 + 21/1024*m^3 + 31/2048*m^4 + 6257/524288*m^5 + ...

Guide to Tables, Math. Tables Other Aids Computation, 3 (1948), Section III, p. 234.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Charles Hermite, Oeuvres.
Charles Hermite, Sur Quelques Développements En Série de la Théorie Des Fonctions Elliptiques , Oeuvres. Vol. 4, Gauthier-Villars, Paris, 1917, p. 477.
Charles Hermite, Annotated scan of a page from the Oeuvres, together with a page from Math. Tables Aids Comp., Vol. 3, 1948 that refers to it.
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 141.
Eric Weisstein's World of Mathematics, Nome
Robert M. Ziff, On Cardy's formula for the critical crossing probability in 2d percolation, J. Phys. A. 28, 1249-1255 (1995).

Cf. A119349 (denominators), A002103 (where there are further references), A005797.

A002639 := proc(n::integer)
    local z;
    # coeftayl(EllipticNome(z),z=0,n) ; # very slow
    taylor(EllipticNome(z),z=0,2*n+1) ;
    convert(%,polynom) ;
    coeff(%,z,2*n) ;
    numer(%) ;
end proc:
seq(A002639(n),n=0..10) ; # R. J. Mathar, Mar 26 2025

Numerator[ CoefficientList[ Series[ EllipticNomeQ[m], {m, 0, 18}], m]] (* Jean-François Alcover, Sep 21 2011 *)

{a(n) = if( n<1, 0, numerator( polcoeff( serreverse( x * prod(k=1, n, (1 + x^k)^(-1)^k, 1 +x * O(x^n))^8), n) / 4^n))}

Edited by Michael Somos, Aug 09 2002

A308835 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)(x/27)^n follows from the series solutions of 2T-d/dx(9(1-x)x*dT/dx)=0.

Original entry on oeis.org

0, 1, 15, 279, 5729, 124554, 2810718, 65114402, 1538182398, 36887880105, 895303119303, 21943398532563, 542209373589501, 13489931811324550, 337599511395854298, 8491805574767197650, 214548940430198454054, 5441921826542937659088, 138512110164878076019560

Offset: 0

Bradley Klee, Jun 27 2019

nonn

Also appears in Ramanujan's theory of elliptic functions, signature 3 (cf. A006480). Almkvist et al. give a real and complex Ansatz for the second-order, ordinary differential equation: T_R = 1 + x*{Z[[x]]}, T_C = T_R*log(x) + x*{Z[[x]]}.

B. C. Berndt, "Ramanujan's Notebooks Part II", Springer, 2012, pages 80-82.

G. Almkvist et al., Generalizations of Clausen's Formula and Algebraic Transformations of Calabi-Yau Differential Equations, Proceedings of the Edinburgh Mathematical Society, 54 (2011), p. 275. [The article is on pages 273-295.]

Cf. A005797, A308836, A308837.

G[nMax_]:=Dot[RecurrenceTable[{Dot[{(3*n-5)^2 (3*n-4)^2 (9*n-4), -18(n - 1)(40 - 197*n + 351*n^2 - 279*n^3 + 81*n^4),81(n - 1)*n^3*(9*n - 13)}, a[n-#] & /@ Reverse[Range[0, 2]]] == 0, a[0] == 0, a[1] == 5/9}, a, {n, 0, nMax}], x^Range[0, nMax]];
qSer[nMax_] := Expand[Times[x, Normal[Series[Exp[ Divide[G[nMax], Hypergeometric2F1[1/3, 2/3, 1, x]]], {x, 0, nMax}]]]];
CoefficientList[(1/k)*qSer[20]/.{x->k*x},x]/.{k->27}

A308836 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)(x/64)^n follows from the series solutions of 3T-d/dx(16(1-x)x*dT/dx)=0.

Original entry on oeis.org

0, 1, 40, 1876, 95072, 5045474, 276107408, 15444602248, 878268335296, 50588345910799, 2944021398570264, 172780225616034252, 10211876493716693664, 607169816926036666486, 36286222314596227018672, 2178246170438379512947864, 131270483744089714062036032

Offset: 0

Bradley Klee, Jun 27 2019

nonn

Also appears in Ramanujan's theory of elliptic functions, signature 4 (cf. A000897). Almkvist et al. give a real and complex Ansatz for the second-order, ordinary differential equation: T_R = 1 + x*{Z[[x]]}, T_C = T_R*log(x) + x*{Z[[x]]}.

B.C. Berndt, "Ramanujan's Notebooks Part II", Springer, 2012, pages 80-82.

G. Almkvist et al., Generalizations of Clausen's Formula and Algebraic Transformations of Calabi-Yau Differential Equations, Proceedings of the Edinburgh Mathematical Society, 54 (2011), p. 275.

Cf. A005797, A308835, A308837.

G[nMax_] := Dot[RecurrenceTable[ {Dot[{(4*n - 7)^2 (4*n - 5)^2 (8*n - 3), -16 (n - 1) (105 - 562*n + 1056*n^2 - 864*n^3 + 256*n^4), 256 (n - 1) n^3 (8*n - 11)},  a[n - #] & /@ Reverse[Range[0, 2]]] == 0, a[0] == 0, a[1] == 5/8}, a, {n, 0, nMax}], x^Range[0, nMax]];
qSer[nMax_] := Expand[Times[x, Normal[ Series[Exp[ Divide[G[nMax], Hypergeometric2F1[1/4, 3/4, 1, x]]], {x, 0, nMax}]]]];
CoefficientList[(1/k)*qSer[20] /. {x -> k*x}, x] /. {k -> 64}

A308837 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)(x/432)^n follows from the series solutions of 5T-d/dx(36(1-x)x*dT/dx)=0.

Original entry on oeis.org

0, 1, 312, 107604, 39073568, 14645965026, 5609733423408, 2182717163349896, 859521859502348352, 341679883727799750159, 136868519056531319862408, 55173969942211048781835468, 22360181278518828446785034976, 9103073677708423854325869548662

Offset: 0

Bradley Klee, Jun 27 2019

nonn

Also appears in Ramanujan's theory of elliptic functions, signature 6 (cf. A113424). Almkvist et al. give a real and complex Ansatz for the second-order, ordinary differential equation: T_R = 1 + x*{Z[[x]]}, T_C = T_R*log(x) + x*{Z[[x]]}.

B.C. Berndt, "Ramanujan's Notebooks Part II", Springer, 2012, pages 80-82.

G. Almkvist et al., Generalizations of Clausen's Formula and Algebraic Transformations of Calabi-Yau Differential Equations, Proceedings of the Edinburgh Mathematical Society, 54 (2011), p. 275.

Cf. A005797, A308835, A308836.

G[nMax_] := Dot[RecurrenceTable[{Dot[{(6*n - 11)^2 (6*n - 7)^2 (18*n - 5), -36 (n - 1) (385 - 2426*n + 4968*n^2 - 4248*n^3 + 1296*n^4), 1296 (n - 1) n^3 (18*n - 23)},
a[n - #] & /@ Reverse[Range[0, 2]]] == 0, a[0] == 0, a[1] == 13/18}, a, {n, 0, nMax}], x^Range[0, nMax]];
qSer[nMax_] := Expand[Times[x,Normal[ Series[Exp[Divide[G[nMax], Hypergeometric2F1[1/6, 5/6, 1, x]]], {x, 0, nMax}]]]];
CoefficientList[(1/k)*qSer[12] /. {x -> k*x}, x] /. {k -> 432}

A078791 Expansion of Auxiliary function L(1-m) / 4 in powers of m / 16.

Original entry on oeis.org

0, 1, 21, 740, 37310, 2460024, 200770416, 19551774528, 2213488134000, 285711909912000, 41419784380740480, 6663725042739448320, 1178209566488368028160, 227096910697908706560000

Offset: 0

Michael Somos, Dec 05 2002

Nome q(m) = x exp(8 * (Sum_{n>0} a(n) * x^n / n!) / (Sum_{n>=0} binomial(2n, n)^2 * x^n)) where x = m / 16.

The Fricke reference on page 2 has equation "(3) Pi i omega = -Pi K'/K = log k^2 - 4 log 2 + F_1(1/2, 1/2; k^2) / F(1/2, 1/2, 1; k^2), wo F_1 und F ..." where F_1 = 8 * Sum_{n>0} a(n) * x^n / n! with x = m / 16 = (k / 4)^2. - Michael Somos, Jul 14 2013

			G.f. = x + 21*x^2 + 740*x^3 + 37310*x^4 + 2460024*x^5 + 200770416*x^6 + 19551774528*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 591.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 9.

G. C. Greubel, Table of n, a(n) for n = 0..300
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012.

Cf. A000984, A005797, A098118.

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Log[ EllipticNomeQ[ 16 x] / x] Hypergeometric2F1[ 1/2, 1/2, 1, 16 x] / 8, {x, 0, n}]]; (* Michael Somos, Jul 14 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Log[ EllipticNomeQ[ 16 x] / x] EllipticK[ 16 x] / (4 Pi), {x, 0, n}]]; (* Michael Somos, Jul 14 2013 *)
a[ n_] := If[ n < 0, 0, n! Binomial[ 2 n, n]^2 Sum[ 1/k, {k, n + 1, 2 n}] / 2]; (* Michael Somos, Jul 14 2013 *)
a[ n_] := If[ n < 0, 0, n! Binomial[ 2 n, n]^2 (HarmonicNumber[2 n] - HarmonicNumber[n]) / 2]; (* Michael Somos, Apr 14 2015 *)

{a(n) = if( n<0, 0, sum( k=1, n, 1 / (2*k - 1) / k) / 4 * (2*n)!^2 / n!^3)};

E.g.f.: L(1-m) = log(16 / m) (K(m) / Pi) - K(1-m) = 4 Sum_{n>0} a(n) (m/16)^n / n!.
2 * a(n) = A098118(n) * A000984(n). - Michael Somos, Apr 14 2015
a(n) ~ log(2) * 2^(4*n - 1/2) * n^n / (sqrt(Pi*n) * exp(n)). - Vaclav Kotesovec, Jul 10 2016

A274344 Coefficients in the expansion of q^(1/2) in odd powers of k/4, where q is the Jacobi nome and k^2 the parameter of elliptic functions. Also coefficients in the expansion of q in odd powers of (1/4)*(1 - k') / (1 + k') with k'^2 the complementary parameter.

Original entry on oeis.org

1, 4, 34, 360, 4239, 53148, 694582, 9348664, 128625067, 1800131564, 25538105486, 366348201176, 5304067812296, 77394671803040, 1136872705730600, 16796605751564320, 249415741237963837

Offset: 1

Wolfdieter Lang, Jun 30 2016

k' is the square root of the complementary parameter of elliptic functions. In the Abramowitz-Stegun (A-St) reference, p. 569, k'^2 is called m_1. The relation between k'^2 and k^2, the parameter (called m in A-St), is k'^2 = 1 - k^2.

The expansion of q in odd powers of (1/4)*(1 - k') / (1 + k') appears in the Kneser reference, p. 218, where it is attributed to L. Lindelöf. It is obtained from the expansion of sqrt(q) in odd powers of k/4, namely q^{1/2} = Sum_{n >= 0} a(n)*(k/4)^(2*n+1), which results from the expansion -Pi*K'/K = log(q) = log(k^2/16) + log(1 + Sum_{n>=1} A005797(n+1)*(k^2/16)^n) = log(k^2/16) + 8*(k^2/16) + 52*(k^2/16)^2 + ... (see A-St, p. 591, 17. 3.21, Kneser, p. 216, Fricke, eq. (4), p. 2, and A227505, A274345/A274346). The fact that a replacement of q by q^2 means a replacement of k by (1 - k')/(1 + k') is used (Landen transformation).

Vaclav Kotesovec, Table of n, a(n) for n = 1..800
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], pp. 569, 591.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012., p. 2, eq. (4).
A. Kneser, Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen, J. reine u. angew. Math. 157, 1927, 209 - 218.

Cf. A005797, A227505, A274345, A274346.

CoefficientList[Series[Sqrt[EllipticNomeQ[16*x]/x], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 07 2019 *)

q^{1/2} = Sum_{n >= 0} a(n)*(k/4)^(2*n+1).

q = Sum_{n >= 0} a(n)*((1/4)*(1 - k')/(1 + k'))^(2*n+1).

A274662 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function sn(u, k) divided by sin(v) in terms of the Jacobi nome q and even powers of 2cos(v), with v = u/((2/Pi)K(k)).

Original entry on oeis.org

1, 0, 1, 0, -3, 1, 0, 4, -5, 1, 0, -3, 13, -7, 1, 0, 6, -25, 26, -9, 1, 0, -12, 43, -70, 43, -11, 1, 0, 8, -70, 157, -147, 64, -13, 1, 0, -3, 109, -315, 408, -264, 89, -15, 1, 0, 13, -155, 582, -984, 872, -429, 118, -17, 1, 0, -18, 201, -1001, 2142, -2464, 1641, -650, 151, -19, 1

Offset: 0

Wolfdieter Lang, Aug 08 2016

The representation of Jacobi's elliptic sn(u, k) function in terms of quotients of theta functions of the variables q (Jacobi nome) and v = u/((2/Pi)*K(k)) with the real quarter period K is
  sn(u, k) = (theta_3(0, q)/theta_2(0, q)) * (theta_1(v, q)/theta_4(v, q)).
  This can be written either in terms of infinite sums or products. (see e.g., Tricomi, p. 176, eq. (3.87), p. 156, eq. (3.51), p. 167, eq. (3.71) with (3.71'), p. 173, eq. (3.81)).
The sums representation involves sin((2*n+1)*v) and cos(2*n*v) functions. Using Chebyshev S and T polynomial (A049310 and A053120) one can write sn(u, k)/sin(v) = Sum_{n >= 0} q^n*Sum_{m = 0..n} T(n, m) * (2*cos(v))^(2*m).
The product representation involves directly (2*cos(v))^2 powers in the q expansion:
  sn(u, k)/sin(v) = Product_{n >= 1} (1 - (q^(2*n)/(1 + q^(2*n))^2)*(2*cos(v))^2) / (1 - (q^(2*n-1)/(1 + q^(2*n-1))^2)*(2*cos(v))^2) = Sum_{n >=0} q^n * Sum_{m = 1..n} T(n, m)*(2*cos(v))^(2*m).
This sn expansion in the v and q variables is used in the scaled phase space coordinate qhat(v, q) of the plane pendulum. See A275790.
An alternative expansion of sn in the variables v and q is given in A274659.
See also the W. Lang link, equations (52) and (53).

			The triangle T(n, m) begins:
n\m 0   1    2    3    4    5    6   7   8 9
0:  1
1:  0   1
2:  0  -3    1
3:  0   4   -5    1
4:  0  -3   13   -7    1
5:  0   6  -25   26   -9    1
6:  0 -12   43  -70   43  -11    1
7:  0   8  -70  157 -147   64  -13   1
8:  0  -3  109 -315  408 -264   89 -15   1
9:  0  13 -155  582 -984  872 -429 118 -17 1
...
row n=10: 0 -18 201 -1001 2142 -2464 1641 -650 151 -19 1
...
n=4: the q^4 term of sn(u, k)/sin(v) is -3*(2*cos(v))^2 + 13*(2*cos(v))^4 - 7*(2*cos(v))^6 + (2*cos(v))^8.
One can check the identity for example for u = 1 and k = sqrt(1/2), belonging to v = 0.8472130848  and q = 0.04321391815 (Maple 10 digits), with the result from Maple's sn function sn(1, sqrt(1/2)) = 0.8030018249 (10 digits). If one takes the expansion up to q^4 inclusive one obtains .8030012888 (10 digits).

F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948.

Wolfdieter Lang, Expansions for phase space coordinates for the plane pendulum

Cf. A002103, A005797, A038534/A056982, A049310, A053120, A274659, A275790.

sn(u, k) = sin(v)*Sum_{n >= 0} q^n*Sum_{m = 0..n} T(n, m)*(2*cos(v))^(2*m), becoming an identity when q, the Jacobi nome, is replaced by exp(-Pi*K'(k)/K(k)) and v by u/((2/Pi)*K(k)) with the real and imaginary quarter periods K' and K, respectively. For the expansions of q = q(k) see A005797 or better A002103 for q = q((1-k^2)^(1/4)), and for (2/Pi)*K(k) see A038534 / A056982.

cp's OEIS Frontend

A002057 Fourth convolution of Catalan numbers: a(n) = 4*binomial(2*n+3,n)/(n+4).

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A005798 Expansion of (theta_2(q)/theta_3(q))^4/16 in powers of q.

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A002639 Numerators of expansion of Jacobi nome q in parameter m.

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A308835 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)*(x/27)^n follows from the series solutions of 2*T-d/dx(9*(1-x)*x*dT/dx)=0.

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A308836 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)*(x/64)^n follows from the series solutions of 3*T-d/dx(16*(1-x)*x*dT/dx)=0.

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A308837 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)*(x/432)^n follows from the series solutions of 5*T-d/dx(36*(1-x)*x*dT/dx)=0.

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A078791 Expansion of Auxiliary function L(1-m) / 4 in powers of m / 16.

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A002057 Fourth convolution of Catalan numbers: a(n) = 4binomial(2n+3,n)/(n+4).

A308835 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)(x/27)^n follows from the series solutions of 2T-d/dx(9(1-x)x*dT/dx)=0.

A308836 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)(x/64)^n follows from the series solutions of 3T-d/dx(16(1-x)x*dT/dx)=0.

A308837 The nome q=exp(T_C/T_R)=Sum_{n>=0} a(n)(x/432)^n follows from the series solutions of 5T-d/dx(36(1-x)x*dT/dx)=0.

A274662 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function sn(u, k) divided by sin(v) in terms of the Jacobi nome q and even powers of 2cos(v), with v = u/((2/Pi)K(k)).