A162010 -id:A162010 - cp's OEIS frontend

A162005 The EG1 triangle.

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).

A004004 a(n) = (3^(2n+1) - 8n - 3)/16.

Original entry on oeis.org

0, 1, 14, 135, 1228, 11069, 99642, 896803, 8071256, 72641337, 653772070, 5883948671, 52955538084, 476599842805, 4289398585298, 38604587267739, 347441285409712, 3126971568687473, 28142744118187326, 253284697063686007, 2279562273573174140, 20516060462158567341

Offset: 0

N. J. A. Sloane, Simon Plouffe

The o.g.f. of this sequence enabled the analysis of A162008, A162009 and A162010. - Johannes W. Meijer, Jun 27 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
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.
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.
Index entries for linear recurrences with constant coefficients, signature (11,-19,9).

From Johannes W. Meijer, Jun 27 2009: (Start)
Equals the second right hand column of triangle A162005 divided by 2.
Cf. A162008, A162009, A162010, A162011 and A162014 [2*(1+3*z)].
(End)

LinearRecurrence[{11, -19, 9}, {0, 1, 14}, 100] (* G. C. Greubel, Jul 06 2016 *)
Table[(3^(2 n + 1) - 8 n - 3)/16, {n, 0, 24}] (* Michael De Vlieger, Jul 08 2016 *)

G.f.: -x*(1+3*x)/(9*x-1)/(x-1)^2. - Simon Plouffe in his 1992 dissertation.
a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3). - Johannes W. Meijer, Jun 27 2009
a(n) = a(n-1) + (3^(2*n-1) - 1)/2. - Lechoslaw Ratajczak, Jul 06 2016
E.g.f.: (-3 - 8*x + 3*exp(8*x))*exp(x)/16. - Ilya Gutkovskiy, Jul 07 2016

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))

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

A322230 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^2S(x,k)^2 = 1, as a triangle of coefficients read by rows.

Original entry on oeis.org

1, 1, 2, 1, 28, 16, 1, 270, 1032, 272, 1, 2456, 36096, 52736, 7936, 1, 22138, 1035088, 4766048, 3646208, 353792, 1, 199284, 27426960, 319830400, 704357760, 330545664, 22368256, 1, 1793606, 702812568, 18598875760, 93989648000, 120536980224, 38188155904, 1903757312, 1, 16142512, 17753262208, 1002968825344, 10324483102720, 28745874079744, 24060789342208, 5488365862912, 209865342976, 1, 145282674, 445736371872, 51882638754240, 1013356176688128, 5416305638467584, 9498855414644736, 5590122715250688, 961530104709120, 29088885112832

Offset: 0

Paul D. Hanna, Dec 14 2018

Equals a row reversal of triangle A325220.
Appears to be a row reversal of EG1 triangle A162005, which has other formulas.
Compare to sn(x,k) = Integral cn(x,k)*dn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions (see triangle A060628).
Compare also to Michael Pawellek's generalized elliptic functions.

			E.g.f.: S(x,k) = x + (2*k^2 + 1)*x^3/3! + (16*k^4 + 28*k^2 + 1)*x^5/5! + (272*k^6 + 1032*k^4 + 270*k^2 + 1)*x^7/7! + (7936*k^8 + 52736*k^6 + 36096*k^4 + 2456*k^2 + 1)*x^9/9! + (353792*k^10 + 3646208*k^8 + 4766048*k^6 + 1035088*k^4 + 22138*k^2 + 1)*x^11/11! + (22368256*k^12 + 330545664*k^10 + 704357760*k^8 + 319830400*k^6 + 27426960*k^4 + 199284*k^2 + 1)*x^13/13! + ...
such that C(x,k)^2 - S(x,k)^2 = 1.
This triangle of coefficients T(n,j) of x^(2*n+1)*k^(2*j)/(2*n+1)! in e.g.f. S(x,k) begins:
1;
1, 2;
1, 28, 16;
1, 270, 1032, 272;
1, 2456, 36096, 52736, 7936;
1, 22138, 1035088, 4766048, 3646208, 353792;
1, 199284, 27426960, 319830400, 704357760, 330545664, 22368256;
1, 1793606, 702812568, 18598875760, 93989648000, 120536980224, 38188155904, 1903757312;
1, 16142512, 17753262208, 1002968825344, 10324483102720, 28745874079744, 24060789342208, 5488365862912, 209865342976; ...
RELATED SERIES.
The related series C(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
C(x,k) = 1 + x^2/2! + (8*k^2 + 1)*x^4/4! + (136*k^4 + 88*k^2 + 1)*x^6/6! + (3968*k^6 + 6240*k^4 + 816*k^2 + 1)*x^8/8! + (176896*k^8 + 513536*k^6 + 195216*k^4 + 7376*k^2 + 1)*x^10/10! + (11184128*k^10 + 51880064*k^8 + 39572864*k^6 + 5352544*k^4 + 66424*k^2 + 1)*x^12/12! + (951878656*k^12 + 6453433344*k^10 + 8258202240*k^8 + 2458228480*k^6 + 139127640*k^4 + 597864*k^2 + 1)*x^14/14! + ...
The related series D(x,k), where D(x,k)^2 - k^2*S(x,k)^2 = 1, starts
D(x,k) = 1 + k^2*x^2/2! + (5*k^4 + 4*k^2)*x^4/4! + (61*k^6 + 148*k^4 + 16*k^2)*x^6/6! + (1385*k^8 + 6744*k^6 + 2832*k^4 + 64*k^2)*x^8/8! + (50521*k^10 + 410456*k^8 + 383856*k^6 + 47936*k^4 + 256*k^2)*x^10/10! + (2702765*k^12 + 32947964*k^10 + 54480944*k^8 + 17142784*k^6 + 780544*k^4 + 1024*k^2)*x^12/12! + (199360981*k^14 + 3402510924*k^12 + 8760740640*k^10 + 5199585280*k^8 + 686711040*k^6 + 12555264*k^4 + 4096*k^2)*x^14/14! + ...

Paul D. Hanna, Table of n, a(n) for n = 0..860 terms of this triangle read by rows 0..40.

Cf. A000182 (diagonal), A162006, A162007, A162008, A162009, A162010.
Cf. A322231 (C), A322232 (D).
Cf. A325220 (row reversal), A162005.

N=10;
{S=x;C=1;D=1; for(i=1,2*N, S = intformal(C*D^2 +O(x^(2*N+1))); C = 1 + intformal(S*D^2); D = 1 + k^2*intformal(S*C*D));}
for(n=0,N, for(j=0,n, print1( (2*n+1)!*polcoeff(polcoeff(S,2*n+1,x),2*j,k),", ")) ;print(""))

E.g.f. S = S(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n+1) * k^(2*j) / (2*n+1)!, along with related series C = C(x,k) and D = D(x,k), satisfies:
(1a) S = Integral C*D^2 dx.
(1b) C = 1 + Integral S*D^2 dx.
(1c) D = 1 + k^2 * Integral S*C*D dx.
(2a) C^2 - S^2 = 1.
(2b) D^2 - k^2*S^2 = 1.
(3a) C + S = exp( Integral D^2 dx ).
(3b) D + k*S = exp( k * Integral C*D dx ).
(4a) S = sinh( Integral D^2 dx ).
(4b) S = sinh( k * Integral C*D dx ) / k.
(4c) C = cosh( Integral D^2 dx ).
(4d) D = cosh( k * Integral C*D dx ).
(5a) d/dx S = C*D^2.
(5b) d/dx C = S*D^2.
(5c) d/dx D = k^2 * S*C*D.
From Paul D. Hanna, Mar 31 2019, Apr 20 2019 (Start):
Given sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions, with i^2 = -1, k' = sqrt(1-k^2), then
(6a) S = -i * sn( i * Integral D dx, k),
(6b) C = cn( i * Integral D dx, k),
(6c) D = dn( i * Integral D dx, k).
(7a) S = sc( Integral D dx, k') = sn(Integral D dx, k')/cn(Integral D dx, k'),
(7b) C = nc( Integral D dx, k') = 1/cn(Integral D dx, k'),
(7c) D = dc( Integral D dx, k') = dn(Integral D dx, k')/cn(Integral D dx, k'). (End)
Row sums equal (2*n+1)!*(2*n)!/(n!^2*4^n) = A079484(n), the product of two consecutive odd double factorials.
Main diagonal equals A000182, the tangent numbers.

cp's OEIS Frontend

A162005 The EG1 triangle.

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A004004 a(n) = (3^(2*n+1) - 8*n - 3)/16.

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

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Maple

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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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A322230 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, as a triangle of coefficients read by rows.

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PARI

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A004004 a(n) = (3^(2n+1) - 8n - 3)/16.

A322230 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^2S(x,k)^2 = 1, as a triangle of coefficients read by rows.