A004004 -id:A004004 - cp's OEIS frontend

A060628 Triangle of coefficients in expansion of elliptic function sn(u) in powers of u and k.

1, 1, 1, 1, 14, 1, 1, 135, 135, 1, 1, 1228, 5478, 1228, 1, 1, 11069, 165826, 165826, 11069, 1, 1, 99642, 4494351, 13180268, 4494351, 99642, 1, 1, 896803, 116294673, 834687179, 834687179, 116294673, 896803, 1, 1, 8071256, 2949965020, 47152124264, 109645021894, 47152124264, 2949965020, 8071256, 1

Offset: 0

Vladeta Jovovic, Apr 13 2001

			sn u = u - (1 + k^2)*u^3/3! + (1 + 14*k^2 + k^4)*u^5/5! - (1 + 135*k^2 + 135*k^4 + k^6)*u^7/7! + ...
The triangle T(n, m) begins:
n\m 0      1         2         3         4         5      6  7
0:  1
2:  1      1
3:  1     14         1
4:  1    135       135         1
5:  1   1228      5478      1228         1
6:  1  11069    165826    165826     11069         1
7:  1  99642   4494351  13180268   4494351     99642      1
8:  1 896803 116294673 834687179 834687179 116294673 896803  1
... reformatted. - _Wolfdieter Lang_, Jul 05 2016

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 526.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.2.24).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.22.1.
A. Cayley, An Elementary Treatise on Elliptic Functions (page images), G. Bell and Sons, London, 1895, p. 56.
F. Clarke, The Taylor Series Coefficients of the Jacobi Elliptic Functions, slides. [broken link]
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.
A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Erster Teil, p. 399 with p. 397.
C. L. Mallows, Letter to N. J. A. Sloane, May 16 1973
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 92.
G. Viennot, Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi, J. Combin. Theory, A 29 (1980), 121-133.
Eric Weisstein's World of Mathematics, Jacobi Elliptic Functions

Row sums give A000182. Diagonals: A004004, A004005, A002753, A032348, A032427, A032428, A032429, A032430, A032431, A032432, A032433.

Maple program from Rostislav Kollman (kollman(AT)dynasig.cz), Nov 05 2009: (Start) The program generates an "all in one" triangle of Taylor coefficients of the Jacobi SN,CN,DN functions.
"SN ", 1 "CN ", 1 "DN ", 1
"SN ", 1, 1 "CN ", 1, 4 "DN ", 4, 1
"SN ", 1, 14, 1 "CN ", 1, 44, 16 "DN ", 16, 44, 1
"SN ", 1, 135, 135, 1 "CN ", 1, 408, 912, 64 "DN ", 64, 912, 408, 1
"SN ", 1, 1228, 5478, 1228, 1 "CN ", 1, 3688, 30768, 15808, 256 "DN ", 256, 15808, 30768, 3688, 1
"SN ", 1, 11069, 165826, 165826, 11069, 1 "CN ", 1, 33212, 870640, 1538560, 259328, 1024 "DN ", 1024, 259328, 1538560, 870640, 33212, 1
#----------------------------------------------------------------
# Taylor series coefficients of Jacobi SN,CN,DN
#----------------------------------------------------------------
n := 6: g := x: for i from 1 to 2*n do g := simplify(y*z*diff(g,x) + x*z*diff(g,y) + x*y*diff(g,z)); if(type(i,odd))then SN := simplify(sort(subs(z = k,subs(y = 1,subs(x = 0,g)))) / k);
# lprint("SN ",SN); lprint("SN ",seq(coeff(SN, k, j),j=0..i-1,2)); else CN := simplify(sort(subs(z = 1,subs(y = 0,subs(x = k,g)))) / k); DN := simplify(sort(subs(z = 0,subs(y = k,subs(x = 1,g)))));
# lprint("CN ",CN); # lprint("DN ",DN); lprint("CN ",seq(coeff(CN, k, j),j=0..i-2,2)); lprint("DN ",seq(coeff(DN, k, j),j=2..i,2)); end; end: (End)
A060628 := proc(n,m) JacobiSN(z,k) ; coeftayl(%,z=0,2*n+1) ; (-1)^n*coeftayl(%,k=0,2*m)*(2*n+1)! ; end proc: # alternative program, R. J. Mathar, Jan 30 2011

maxn = 8; se = Series[ JacobiSN[u, m], {u, 0, 2*maxn + 1 }]; cc = Partition[ CoefficientList[se, u], 2][[All, 2]]; Flatten[ (CoefficientList[#, m] & /@ cc)* Table[(-1)^n*(2*n + 1)!, {n, 0, maxn}]] (* Jean-François Alcover, Sep 21 2011 *)

Sum_{n>=0} Sum_{k=0..n} (-1)^n*T(n, k)*y^(2*k)*x^(2*n+1)/(2*n+1)! = JacobiSN(x, y).
JacobiSN(x, y) = 1*x + (-1/6 - (1/6)*y^2)*x^3 + (1/120 + (7/60)*y^2 + (1/120)*y^4)*x^5 + (-1/5040 - (3/112)*y^4 - (3/112)*y^2 - (1/5040)*y^6)*x^7 + (1/362880 + (307/90720)*y^6 + (913/60480)*y^4 + (307/90720)*y^2 + (1/362880)*y^8)*x^9 + O(x^11).
From Peter Bala, Aug 23 2011: (Start)
Let f(x) = sqrt((1-x^2)*(1-k^2*x^2)).
Define the nested derivative D^n[f](x) by means of the recursion D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0.
Then the coefficient polynomial R(2*n+1,k) of u^(2*n+1)/(2*n+1)! is given by R(2*n+1,k) = D^(2*n)[f](0) - apply [Dominici, Theorem 4.1].
See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x).
(End)
sn(u|k^2) = Sum_{n>=0} a_n(k^2)*u^(2*n+1)/(2*n+1)!. For the recurrence of the row polynomials a_n(k^2) = Sum_{m=0..n} (-1)^n*T(n, m)*k^(2*m), see the Fricke reference. - Wolfdieter Lang, Jul 05 2016

A162005 The EG1 triangle.

Original entry on oeis.org

1, 2, 1, 16, 28, 1, 272, 1032, 270, 1, 7936, 52736, 36096, 2456, 1, 353792, 3646208, 4766048, 1035088, 22138, 1, 22368256, 330545664, 704357760, 319830400, 27426960, 199284, 1, 1903757312, 38188155904, 120536980224, 93989648000

Offset: 1

Johannes W. Meijer, Jun 27 2009, Jul 02 2009, Aug 31 2009

We define the EG1 matrix by EG1[2m-1,1] = 2*eta(2m-1) and the recurrence relation EG1[2m-1,n] = EG1[2m-1,n-1] - EG1[2m-3,n-1]/(n-1)^2 for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .., with eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function. For the EG2[2m,n] coefficients see A008955.
The n-th term of the row coefficients EG1[1-2*m,n] for m = 1, 2, .., can be generated with REG1(1-2*m,n) = (-1)^(m+1)*2^(1-m)*ECGP(1-2*m, n)*(1/n)*4^(-n)*(2*n)!/((n-1)!)^2 . For information about the ECGP polynomials see A094665 and the examples below.
We define the o.g.f.s. of the REG1(1-2*m,n) by GFREG1(z,1-2*m) = sum(REG1(1-2*m,n)* z^(n-1), n=1..infinity) for m = 1, 2, .., with GFREG1(z,1-2*m) = (-1)^(m+1)* RG(z,1-2*m)/ (2^(2*m-1)*(1-z)^((2*m+1)/2)). The RG(z,1-2m) polynomials led to the EG1 triangle.
We used the coefficients of the A156919 and A094665 triangles to determine those of the EG1 triangle, see the Maple program. The A156919 triangle gives information about the sums SF(p) = sum(n^(p-1)*4^(-n)*z^(n-1)*(2*n)!/((n-1)!)^2, n=1..infinity) for p= 0, 1, 2, .. .
Contribution from Johannes W. Meijer, Nov 23 2009: (Start)
The EG1 matrix is related to the ED2 array A167560 because sum(EG1(2*m-1,n)*z^(2*m-1), m=1..infinity) = ((2*n-1)!/(4^(n-1)*(n-1)!^2))*int(sinh(y*(2*z))/cosh(y)^(2*n),y=0..infinity).
(End)
Appears to equal triangle A322230 with rows read in reverse order. Triangle A322230 describes the e.g.f. S(x,k) = Integral C(x,k)*D(x,k)^2 dx, such that C(x,k)^2 - S(x,k)^2 = 1, and D(x,k)^2 - k^2*S(x,k)^2 = 1. - Paul D. Hanna, Dec 22 2018
Appears to equal triangle A325220, which has e.g.f. S(x,k) = -i * sn( i * Integral C(x,k) dx, k) such that C(x,k) = cn( i * Integral C(x,k) dx, k), where sn(x,k) and cn(x,k) are Jacobi Elliptic functions. - Paul D. Hanna, Apr 13 2019

			The first few rows of the EG1 triangle are :
[1]
[2, 1]
[16, 28, 1]
[272, 1032, 270, 1]
The first few RG(z,1-2*m) polynomials are:
RG(z,-1) = 1
RG(z,-3) = 2+z
RG(z,-5) = 16+28*z+z^2
RG(z,-7) = 272+1032*z+270*z^2+z^3
The first few GFREG1(z,1-2*m) are:
GFREG1(z,-1) = (1)*(1)/(2*(1-z)^(3/2))
GFREG1(z,-3) = (-1)*(2+z)/(2^3*(1-z)^(5/2))
GFREG1(z,-5) = (1)*(16+28*z+z^2)/( 2^5*(1-z)^(7/2))
GFREG1(z,-7) = (-1)*(272+1032*z+270*z^2+z^3)/(2^7*(1-z)^(9/2))
The first few REG1(1-2*m,n) are:
REG1(-1,n) = (1/1)*(1)*(1/n)*4^(-n)*(2*n)!/(n-1)!^2
REG1(-3,n) = (-1/2)*(n) *(1/n)*4^(-n)*(2*n)!/(n-1)!^2
REG1(-5,n) = (1/4) *(n+3*n^2) *(1/n)*4^(-n)*(2*n)!/(n-1)!^2
REG1(-7,n) = (-1/8)*(4*n+15*n^2+15*n^3) *(1/n)*4^(-n)*(2*n)!/(n-1)!^2
The first few ECGP(1-2*m,n) polynomials are:
ECGP(-1,n) = 1
ECGP(-3,n) = n
ECGP(-5,n) = n+3*n^2
ECGP(-7,n) = 4*n+15*n^2+15*n^3

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.

A079484 equals the row sums.
A000182 (ZAG numbers), A162006 and A162007 equal the first three left hand columns.
A000012, A004004 (2x), A162008, A162009 and A162010 equal the first five right hand columns.
Related to A094665, A083061 and A156919 (DEF triangle).
Cf. A161198 [(1-x)^((-1-2*n)/2)], A008955 (EG2[2m, n])
Cf. A167560 (ED2 array).
Cf. A322230 (reversed rows), A325220.

nmax:=7; mmax := nmax: imax := nmax: T1(0, x) := 1: T1(0, x+1) := 1: for i from 1 to imax do T1(i, x) := expand((2*x+1) * (x+1)*T1(i-1, x+1)-2*x^2*T1(i-1, x)): dx := degree(T1(i, x)): for k from 0 to dx do c(k) := coeff(T1(i, x), x, k) od: T1(i, x+1) := sum(c(j1)*(x+1)^(j1), j1=0..dx): od: for i from 0 to imax do for j from 0 to i do A083061(i, j) := coeff(T1(i, x), x, j) od: od: for n from 0 to nmax do for k from 0 to n do A094665(n+1, k+1) := A083061(n, k) od: od: A094665(0, 0) := 1: for n from 1 to nmax do A094665(n, 0) := 0 od: for m from 1 to mmax do A156919(0, m) := 0 end do: for n from 0 to nmax do A156919(n, 0) := 2^n end do: for n from 1 to nmax do for m from 1 to mmax do A156919(n, m) := (2*m+2)*A156919(n-1, m) + (2*n-2*m+1)*A156919(n-1, m-1) end do end do: for n from 0 to nmax do SF(n) := sum(A156919(n, k1)*z^k1, k1=0..n)/(2^(n+1)*(1-z)^((2*n+3)/2)) od: GFREG1(z, -1) := A156919(0, 0)*A094665 (0, 0) / (2*(1-z)^(3/2)): for m from 2 to nmax do GFREG1(z, 1-2*m) := simplify((-1)^(m+1)*2^(1-m)* sum(A094665(m-1, k2)*SF(k2), k2=1..m-1)) od: for m from 1 to mmax do g(m) := sort((numer ((-1)^(m+1)* GFREG1(z, 1-2*m))), ascending) od: for n from 1 to nmax do for m from 1 to n do a(n, m) := abs(coeff(g(n), z, m-1)) od: od: seq(seq(a(n, m), m=1..n), n=1..nmax);
# Maple program edited by Johannes W. Meijer, Sep 25 2012

A different form of the recurrence relation is EG1[1-2*m,n] = (EG1[3-2*m,n]-EG1[3-2*m,n+1])* (n^2) for m = 2, 3, .., with EG1[ -1,n] = (1/n)*4^(-n)*((2*n)!/(n-1)!^2).

A162008 Third right hand column of the EG1 triangle A162005.

Original entry on oeis.org

16, 1032, 36096, 1035088, 27426960, 702812568, 17753262208, 445736371872, 11162877175440, 279268061007400, 6983654996144256, 174610650719469552, 4365455001524490256, 109138210900706764728, 2728473030627812279040

Offset: 1

Johannes W. Meijer, Jun 27 2009

Index entries for linear recurrences with constant coefficients, signature (46,-663,3748,-7711,6606,-2025).

Third right hand column of the EG1 triangle A162005.
Other right hand columns are A004004 (2x), A162009 and A162010.
Cf. A162011 and A162014.

a(n) = (64*n^2+112*n+30-432*n*9^n-405*9^n+375*25^n)/128.
a(n) = 46*a(n-1)-663*a(n-2)+3748*a(n-3)-7711*a(n-4)+6606*a(n-5)-2025*a(n-6).
G.f.: (16+296*z-768*z^2-1080*z^3)/((1-z)^3*(1-9*z)^2*(1-25*z)).

A162009 Fourth right hand column of the EG1 triangle A162005.

Original entry on oeis.org

272, 52736, 4766048, 319830400, 18598875760, 1002968825344, 51882638754240, 2621627565515520, 130715075544000720, 6468157990602644480, 318685706549526508832, 15663443488266952501376, 768809642314801857986608

Offset: 1

Johannes W. Meijer, Jun 27 2009

Fourth right hand column of the EG1 triangle A162005.
Other right hand columns are A004004 (2x), A162008 and A162010.
Cf. A162011 and A162014.

a(n) = (252105*49^n-525+76545*9^n-328125*25^n-225000*25^n*n-2320*n+134136*9^n*n-2112*n^2+46656*9^n*n^2-512*n^3)/3072
a(n) = 130*a(n-1)-6501*a(n-2)+163160*a(n-3)-2236466*a(n-4)+17123340*a(n-5)-71497186*a(n-6)+154127320*a(n-7)-174334221*a(n-8)+98986050*a(n-9)-22325625*a(n-10)
GF(z) = (272+17376*z-321360*z^2-1298624*z^3+8914800*z^4-11262240*z^5-10206000*z^6)/((1-z)^4*(1-9*z)^3*(1-25*z)^2*(1-49*z))

A162010 Fifth right hand column of the EG1 triangle A162005.

Original entry on oeis.org

7936, 3646208, 704357760, 93989648000, 10324483102720, 1013356176688128, 92857038223998720, 8148225153293502720, 695389790665420312320, 58282750219059501633280, 4827428305286309709508736

Offset: 1

Johannes W. Meijer, Jun 27 2009

Fifth right hand column of the EG1 triangle A162005.
Other right hand columns are A004004 (2x), A162008 and A162009.
Cf. A162011 and A162014.

a(n) = (13230+75008*n^2+65184*n-778248135*49^n+30720*n^3+ 502211745*81^n+ 295312500*25^n-26034048*9^n*n^2-43600032*9^n*n-395300640*49^n*n-19289340*9^n+ 4096*n^4+ 352500000*25^n*n+90000000*25^n*n^2-4478976*9^n*n^3)/98304
a(n)= 295*a(n-1)-36729*a(n-2)+2549775*a(n-3)-109746165*a(n-4)+3080128275*a(n-5)-57713313405*a(n-6)+727045264875*a(n-7)-6122436806115*a(n-8)+33837597147925*a(n-9)-119061300168619*a(n-10)+257794693911405*a(n-11)-339251103039591*a(n-12)+264193039731825*a(n-13)-112000136889375*a(n-14)+19937341265625*a(n-15)
GF(z) = (7936+1305088*z-79792256*z^2-109331968*z^3+41828672000*z^4-460917924352*z^5+238697445120*z^6+5066784271872*z^7-14723693948160*z^8+12172737024000*z^9+8101522800000*z^10)/(1-z)^5/(1-9*z)^4/(1-25*z)^3/(1-49*z)^2/(1-81*z)

A162011 A sequence related to the recurrence relations of the right hand columns of the EG1 triangle A162005.

Original entry on oeis.org

1, -1, 1, -11, 19, -9, 1, -46, 663, -3748, 7711, -6606, 2025, 1, -130, 6501, -163160, 2236466, -17123340, 71497186, -154127320, 174334221, -98986050, 22325625, 1, -295, 36729, -2549775, 109746165, -3080128275, 57713313405, -727045264875

Offset: 1

Johannes W. Meijer, Jun 27 2009

The recurrence relation RR(n) = 0 of the n-th right hand column can be found with RR(n) = expand( product((1-(2*k-1)^2*z)^(n-k+1),k=1..n),z) = 0 and replacing z^p by a(n-p).

The polynomials in the numerators of the generating functions GF(z) of the coefficients that precede the a(n), a(n-1), a(n-2) and a(n-3) sequences, see A000012, A006324, A162012 and A162013, are symmetrical. This phenomenon leads to the sequence [1, 1, 6, 1, 19, 492, 1218, 492, 19 , 9, 3631, 115138, 718465, 1282314, 718465, 115138, 3631, 9].

			The recurrence relations for the first few right hand columns:
n = 1: a(n) = 1*a(n-1)
n = 2: a(n) = 11*a(n-1)-19*a(n-2)+9*a(n-3)
n = 3: a(n) = 46*a(n-1)-663*a(n-2)+3748*a(n-3)-7711*a(n-4)+6606*a(n-5)-2025*a(n-6)
n = 4: a(n) = 130*a(n-1)-6501*a(n-2)+163160*a(n-3)-2236466*a(n-4)+17123340*a(n-5)-71497186*a(n-6)+154127320*a(n-7)-174334221*a(n-8)+98986050*a(n-9)-22325625*a(n-10)

A000012, A004004 (2x), A162008, A162009 and A162010 are the first five right hand columns of EG1 triangle A162005.
A000124 (the Lazy Caterer's sequence) gives the number of terms of the RR(n).
A006324, A162012 and A162013 equal the absolute values of the coefficients that precede the a(n-1), a(n-2) and a(n-3) factors of the RR(n).

nmax:=5; for n from 1 to nmax do RR(n) := expand(product((1-(2*k-1)^2*z)^(n-k+1), k=1..n), z) od: T:=1: for n from 1 to nmax do for m from 0 to(n)*(n+1)/2 do a(T):= coeff(RR(n), z, m): T:=T+1 od: od: seq(a(k), k=1..T-1);

RR(n) = expand( product((1-(2*k-1)^2*z)^(n-k+1),k=1..n),z) with n = 1, 2, 3, .. . The coefficients of these polynomials lead to the sequence given above.

A162014 Sequence related to the o.g.f.s. of the right hand columns of the EG1 triangle A162005.

Original entry on oeis.org

1, 8, -1536, -14155776, 10436770529280, 923378661099307008000, -13724698564186788948502118400000, -45695540009113634492156662349750599680000000

Offset: 1

Johannes W. Meijer, Jun 27 2009

The a(n) are the sums of the coefficients of the polynomials that appear in the numerators of the o.g.f.s. of the right hand columns of the EG1 triangle A162005, see the examples.

			The polynomials in the numerators of the first few o.g.f.s are:
numer(GF(1)) = 1
numer(GF(2)) = 2+6*z
numer(GF(3)) = 16+296*z-768*z^2-1080*z^3
numer(GF(4)) = 272+17376*z-321360*z^2-1298624*z^3+8914800*z^4-11262240*z^5-10206000*z^6
numer(GF(5)) = 7936 + 1305088*z - 79792256*z^2 - 109331968*z^3 + 41828672000*z^4-460917924352*z^5 + 238697445120*z^6 + 5066784271872*z^7 - 14723693948160*z^8+ 12172737024000*z^9 + 8101522800000*z^10

A000012, A004004 (2x), A162008, A162009 and A162010 are the first five right hand columns of the EG1 triangle A162005.

Cf. A055209 and A059332.

a(n) = (-1)^( (n^2+n-2)/2)*4^((n-1)*n/2)*n!*product(k!, k=0..n-1)^2

A002753 Coefficients of elliptic function sn.

Original entry on oeis.org

1, 1, 14, 135, 5478, 165826, 13180268, 834687179, 109645021894, 11966116940238, 2347836365864484, 393938089395885894, 107947764316226205276, 25835579116799316507780

Offset: 0

N. J. A. Sloane

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, Eq. 16.22.1 and 16.22.2.
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
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. 575, Eq. 16.22.1 and 16.22.2.
A. Cayley, An Elementary Treatise on Elliptic Functions (page images), G. Bell and Sons, London, 1895, p. 56.
C. L. Mallows, Letter to N. J. A. Sloane, May 16 1973.
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 92.

Cf. A006089, A060628.

lim = 14; se = Series[ JacobiSN[u, m], {u, 0, 2 lim -1}]; a[n_] := (-1)^n*Coefficient[ SeriesCoefficient[se, 2n + 1]*(2n + 1)!, m^Floor[n/2]]; a[0] = a[1] = 1; Table[a[n], {n, 0, lim-1}] (* Jean-François Alcover, Sep 21 2011 *)

a(n) ~ 2^(4*n + 13/2) * n^(2*n + 1) / (exp(2*n) * Pi^(2*n + 2)). - Vaclav Kotesovec, Apr 10 2018

More terms from Paolo Dominici (pl.dm(AT)libero.it) using formulas 16.22.1 and 16.22.2 of Abramowitz and Stegun's Handbook of Mathematical Functions

A004005 Coefficients of elliptic function sn.

Original entry on oeis.org

1, 135, 5478, 165826, 4494351, 116294673, 2949965020, 74197080276, 1859539731885, 46535238000235, 1163848723925346, 29100851707716150, 727566807977891803, 18189614152200873621, 454744658216502193656

Offset: 2

N. J. A. Sloane, Simon Plouffe

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(5.2.24).
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 = 2..100
A. Cayley, An Elementary Treatise on Elliptic Functions (page images), G. Bell and Sons, London, 1895, p. 56.
A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.
C. L. Mallows, Letter to N. J. A. Sloane, May 16 1973
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 92.
G. Viennot, Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi, J. Combin. Theory, A 29 (1980), 121-133.

Leading terms in rows of triangle in A060628.

A004005:=-(-1-89*z+69*z**2+405*z**3)/(-1+25*z)/(9*z-1)**2/(z-1)**3; # Conjectured by Simon Plouffe in his 1992 dissertation.
A004005 := proc(n) A060628(n,2) ; end proc: seq(A004005(n),n=2..40) ; # R. J. Mathar, Jan 30 2011

maxn = 16; se = Series[JacobiSN[u, m], {u, 0, 2*maxn+1}]; cc = Partition[CoefficientList[se, u], 2][[All, 2]]; cc2 = (CoefficientList[#, m] & /@ cc)*Table[(-1)^n*(2*n+1)!, {n, 0, maxn}]; Table[cc2[[n+1, n-1]], {n, 2, maxn}](* Jean-François Alcover, Feb 17 2012 *)

a(n) = (5^(2*n+1) - (8*n-4)*3^(2*n+1) + 32*n^2 - 32*n -17)/256. - Vaclav Kotesovec after Fransen, Jul 30 2013

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

cp's OEIS Frontend

A060628 Triangle of coefficients in expansion of elliptic function sn(u) in powers of u and k.

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A162005 The EG1 triangle.

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A162008 Third right hand column of the EG1 triangle A162005.

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A162009 Fourth right hand column of the EG1 triangle A162005.

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A162010 Fifth right hand column of the EG1 triangle A162005.

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A162011 A sequence related to the recurrence relations of the right hand columns of the EG1 triangle A162005.

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A162014 Sequence related to the o.g.f.s. of the right hand columns of the EG1 triangle A162005.

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A002753 Coefficients of elliptic function sn.

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A004005 Coefficients of elliptic function sn.

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