cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A153300 Coefficient of x^(4n)/(4n)! in the Maclaurin expansion of cm4(x), which is a generalization of the Dixon elliptic function cm(x,0) defined by A104134.

Original entry on oeis.org

1, 6, 2268, 7434504, 95227613712, 3354162536029536, 264444869673131894208, 40740588107524550752746624, 11136881432872615930509713801472, 5026062205760019668688216299061782016
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2009

Keywords

Comments

Equals column 0 of triangle A357801.

Examples

			G.f.: cm4(x) = 1 + 6*x^4/4! + 2268*x^8/8! + 7434504*x^12/12! + 95227613712*x^16/16! +...
sm4(x) = x + 18*x^5/5! + 14364*x^9/9! + 70203672*x^13/13! + 1192064637456*x^17/17! +...
These functions satisfy: cm4(x)^4 - sm4(x)^4 = 1 where:
cm4(x)^4 = 1 + 24*x^4/4! + 24192*x^8/8! + 140507136*x^12/12! + 2716743794688*x^16/16 +...
RELATED EXPANSIONS:
cm4(x)^2 = 1 + 12*x^4/4! + 7056*x^8/8! + 28340928*x^12/12! + 419025809664*x^16/16! +...
sm4(x)^2 = 2*x^2/2! + 216*x^6/6! + 368928*x^10/10! + 3000945024*x^14/14! +...
cm4(x)^3 = 1 + 18*x^4/4! + 14364*x^8/8! + 70203672*x^12/12! + 1192064637456*x^16/16! +...
sm4(x)^3 = 6*x^3/3! + 2268*x^7/7! + 7434504*x^11/11! + 95227613712*x^15/15! +...
DERIVATIVES:
d/dx cm4(x) = sm4(x)^3 ;
d^2/dx^2 cm4(x) = 3*cm4(x)^3*sm4(x)^2 ;
d^3/dx^3 cm4(x) = 6*cm4(x)^6*sm4(x) + 9*cm4(x)^2*sm4(x)^5 ;
d^4/dx^4 cm4(x) = 6*cm4(x)^9 + 81*cm4(x)^5*sm4(x)^4 + 18*cm4(x)*sm4(x)^8 ;...

Links

Crossrefs

Cf. A104134; A153301, A153302 (cm4(x)^2 + sm4(x)^2).
Cf. A153303 (cm4(x)^4), A357801.

Programs

  • Mathematica
    With[{n = 9}, CoefficientList[Series[JacobiDN[Sqrt[2] x^(1/4), 1/2]/Sqrt[JacobiCN[Sqrt[2] x^(1/4), 1/2]], {x, 0, n}], x] Table[(4 k)!, {k, 0, n}]] (* Jan Mangaldan, Jan 04 2017 *)
  • PARI
    {a(n)=local(A);if(n<0,0,A=x*O(x);for(i=0,n,A=1+intformal(intformal(A^3)^3));n=4*n;n!*polcoeff(A,n))}

Formula

Define sm4(x)^4 = cm4(x)^4 - 1, where sm4(x) is the g.f. of A153301, then:
d/dx cm4(x) = sm4(x)^3 ;
d/dx sm4(x) = cm4(x)^3 .
a(n) ~ 2^(14*n + 11/4) * Gamma(3/4)^(8*n+1) * n^(4*n) / (exp(4*n) * Pi^(6*n + 3/4)). - Vaclav Kotesovec, Oct 06 2019

A357800 Coefficients T(n,k) of x^(4*n+1)*r^(4*k)/(4*n+1)! in power series S(x,r) = Integral C(x,r)^3 * D(x,r)^3 dx such that C(x,r)^4 - S(x,r)^4 = 1 and D(x,r)^4 - r^4*S(x,r)^4 = 1, as a symmetric triangle read by rows.

Original entry on oeis.org

1, 18, 18, 14364, 58968, 14364, 70203672, 671650056, 671650056, 70203672, 1192064637456, 20707300240704, 47530354598496, 20707300240704, 1192064637456, 52269828456672288, 1437626817559769760, 5941554215913771840, 5941554215913771840, 1437626817559769760, 52269828456672288, 4930307288899134335424, 197041019249105562351744, 1283341580573615116868160, 2308585363008068715943680
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2022

Keywords

Examples

			E.g.f.: S(x,r) = Sum_{n>=0} T(n,k) * x^(4*n+1) * r^(4*k) / (4*n+1)! begins:
S(x,r) = x + (18 + 18*r^4)*x^5/5! + (14364 + 58968*r^4 + 14364*r^8)*x^9/9! + (70203672 + 671650056*r^4 + 671650056*r^8 + 70203672*r^12)*x^13/13! + (1192064637456 + 20707300240704*r^4 + 47530354598496*r^8 + 20707300240704*r^12 + 1192064637456*r^16)*x^17/17! + (52269828456672288 + 1437626817559769760*r^4 + 5941554215913771840*r^8 + 5941554215913771840*r^12 + 1437626817559769760*r^16 + 52269828456672288*r^20)*x^21/21! + (4930307288899134335424 + 197041019249105562351744*r^4 + 1283341580573615116868160*r^8 + 2308585363008068715943680*r^12 + 1283341580573615116868160*r^16 + 197041019249105562351744*r^20 + 4930307288899134335424*r^24)*x^25/25! + ...
where S(x,r) = Integral C(x,r)^3 * D(x,r)^3 dx.
TRIANGLE.
This triangle of coefficients T(n,k) of x^(4*n+1) * r^(4*k) / (4*n+1)! in S(x,r) for n >= 0, k = 0..n, begins:
n = 0: [1];
n = 1: [18, 18];
n = 2: [14364, 58968, 14364];
n = 3: [70203672, 671650056, 671650056, 70203672];
n = 4: [1192064637456, 20707300240704, 47530354598496, 20707300240704, 1192064637456];
n = 5: [52269828456672288, 1437626817559769760, 5941554215913771840, 5941554215913771840, 1437626817559769760, 52269828456672288];
n = 6: [4930307288899134335424, 197041019249105562351744, 1283341580573615116868160, 2308585363008068715943680, 1283341580573615116868160, 197041019249105562351744, 4930307288899134335424]; ...
in which both column 0 and the main diagonal equals A153301.
RELATED SERIES.
C(x,r) = 1 + 6*x^4/4! + (2268 + 6048*r^4)*x^8/8! + (7434504 + 56282688*r^4 + 35126784*r^8)*x^12/12! + (95227613712 + 1409371197696*r^4 + 2514356038656*r^8 + 679185948672*r^12)*x^16/16! + (3354162536029536 + 81696140755536384*r^4 + 284770675495950336*r^8 + 220415417637617664*r^12 + 33022883487154176*r^16)*x^20/20! + (264444869673131894208 + 9583398717725834749440*r^4 + 54913653475645427527680*r^8 + 83079959422282198548480*r^12 + 35701050229143616880640*r^16 + 3393656235362623684608*r^20)*x^24/24! + ...
where C(x,r)^4 - S(x,r)^4 = 1.
D(x,r) = 1 + 6*r^4*x^4/4! + (6048*r^4 + 2268*r^8)*x^8/8! + (35126784*r^4 + 56282688*r^8 + 7434504*r^12)*x^12/12! + (679185948672*r^4 + 2514356038656*r^8 + 1409371197696*r^12 + 95227613712*r^16)*x^16/16! + (33022883487154176*r^4 + 220415417637617664*r^8 + 284770675495950336*r^12 + 81696140755536384*r^16 + 3354162536029536*r^20)*x^20/20! + (3393656235362623684608*r^4 + 35701050229143616880640*r^8 + 83079959422282198548480*r^12 + 54913653475645427527680*r^16 + 9583398717725834749440*r^20 + 264444869673131894208*r^24)*x^24/24! +
where D(x,r)^4 - r^4 * S(x,r)^4 = 1.

Links

Crossrefs

Cf. A153301 (column 0), A357804 (row sums), A357801 (C(x,r)), A357802 (D(x,r)).
Cf. A357540.

Programs

  • PARI
    {T(n, k) = my(S=x, C=1, D=1); for(i=0, n,
S = intformal( C^3*D^3 +O(x^(4*n+4)));
C = 1 + intformal( S^3*D^3);
D = 1 + r^4*intformal( S^3*C^3); );
(4*n+1)!*polcoeff( polcoeff(S, 4*n+1, x), 4*k, r)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))
  • PARI
    /* Using Series Reversion (faster) */
{T(n, k) = my(S = serreverse( intformal( 1/((1 + x^4)^3*(1 + r^4*x^4)^3 +O(x^(4*n+4)) )^(1/4) )) );
(4*n+1)!*polcoeff( polcoeff(S, 4*n+1, x), 4*k, r)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))

Formula

Generating function S(x,r) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(4*n+1) * r^(4*k) / (4*n+1)! and related functions C(x,r) and D(x,r) satisfy the following formulas.
For brevity, some formulas here will use S = S(x,r), C = C(x,r), and D = D(x,r).
(1.a) C(x,r)^4 - S(x,r)^4 = 1.
(1.b) D(x,r)^4 - r^4 * S(x,r)^4 = 1.
(1.c) D(x,r)^4 - r^4 * C(x,r)^4 = 1 - r^4.
Integral formulas.
(2.a) S(x,r) = Integral C(x,r)^3 * D(x,r)^3 dx.
(2.b) C(x,r) = 1 + Integral S(x,r)^3 * D(x,r)^3 dx.
(2.c) D(x,r) = 1 + r^4 * Integral S(x,r)^3 * C(x,r)^3 dx.
(2.d) S(x,r)^4 = Integral 4 * S(x,r)^3 * C(x,r)^3 * D(x,r)^3 dx.
Derivatives.
(3.a) d/dx S(x,r) = C(x,r)^3 * D(x,r)^3.
(3.b) d/dx C(x,r) = S(x,r)^3 * D(x,r)^3.
(3.c) d/dx D(x,r) = r^4 * S(x,r)^3 * C(x,r)^3.
Exponential formulas.
(4.a) C + S = exp( Integral (C^2 - C*S + S^2) * D^3 dx ).
(4.b) D + r*S = exp( r * Integral (D^2 - r*D*S + r^2*S^2) * C^3 dx ).
(4.c) C - S = exp( -Integral (C^2 + C*S + S^2) * D^3 dx ).
(4.d) D - r*S = exp( -r * Integral (D^2 + r*D*S + r^2*S^2) * C^3 dx ).
(5.a) C^2 + S^2 = exp( 2 * Integral S*C * D^3 dx ).
(5.b) D^2 + r^2*S^2 = exp( 2*r^2 * Integral S*D * C^3 dx ).
(5.c) C^2 - S^2 = exp( -2 * Integral S*C * D^3 dx ).
(5.d) D^2 - r^2*S^2 = exp( -2*r^2 * Integral S*D * C^3 dx ).
Hyperbolic functions.
(6.a) C = sqrt(C^2 - S^2) * cosh( Integral (C^2 + S^2) * D^3 dx ).
(6.b) S = sqrt(C^2 - S^2) * sinh( Integral (C^2 + S^2) * D^3 dx ).
(6.c) D = sqrt(D^2 - r^2*S^2) * cosh( r * Integral (D^2 + r^2*S^2) * C^3 dx ).
(6.d) r*S = sqrt(D^2 - r^2*S^2) * sinh( r * Integral (D^2 + r^2*S^2) * C^3 dx ).
(7.a) C^2 = cosh( 2 * Integral S*C * D^3 dx ).
(7.b) S^2 = sinh( 2 * Integral S*C * D^3 dx ).
(7.c) D^2 = cosh( 2*r^2 * Integral S*D * C^3 dx ).
(7.d) r^2*S^2 = sinh( 2*r^2 * Integral S*D * C^3 dx ).
Other formulas.
(8) S(x,r) = Series_Reversion( Integral ( (1 + x^4)^3 * (1 + r^4*x^4)^3 )^(-1/4) dx ).
(9.a) T(n,0) = T(n,n) = A153301(n).
(9.b) Sum_{k=0..n} T(n,k) = A357804(n), for n >= 0.

A357541 Coefficients T(n,k) of x^(3*n)*r^(3*k)/(3*n)! in power series C(x,r) = 1 + Integral S(x,r)^2 * D(x,r)^2 dx such that C(x,r)^3 - S(x,r)^3 = 1 and D(x,r)^3 - r^3*S(x,r)^3 = 1, as a triangle read by rows.

Original entry on oeis.org

1, 2, 0, 40, 120, 0, 3680, 37440, 21600, 0, 880000, 20592000, 38966400, 8553600, 0, 435776000, 19269888000, 79491456000, 57708288000, 6329664000, 0, 386949376000, 28748332800000, 213892766208000, 335872728576000, 123646051584000, 7852204800000, 0, 560034421760000, 64544356546560000, 774705298498560000, 2169194182594560000, 1730103155573760000, 374841224017920000, 15132769090560000, 0
Offset: 0

Views

Author

Paul D. Hanna, Oct 09 2022

Keywords

Comments

Related to Dixon elliptic function cm(x,0) (cf. A104134).
Equals a row reversal of triangle A357542, which describes the related function D(x,r).

Examples

			E.g.f.: C(x,r) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(3*n) * r^(3*k) / (3*n)! begins:
C(x,r) = 1 + Integral S(x,r)^2 * D(x,r)^2 dx = 1 + 2*x^3/3! + (40 + 120*r^3)*x^6/6! + (3680 + 37440*r^3 + 21600*r^6)*x^9/9! + (880000 + 20592000*r^3 + 38966400*r^6 + 8553600*r^9)*x^12/12! + (435776000 + 19269888000*r^3 + 79491456000*r^6 + 57708288000*r^9 + 6329664000*r^12)*x^15/15! + (386949376000 + 28748332800000*r^3 + 213892766208000*r^6 + 335872728576000*r^9 + 123646051584000*r^12 + 7852204800000*r^15)*x^18/18! + (560034421760000 + 64544356546560000*r^3 + 774705298498560000*r^6 + 2169194182594560000*r^9 + 1730103155573760000*r^12 + 374841224017920000*r^15 + 15132769090560000*r^18)*x^21/21! + ...
This table of coefficients T(n,k) of x^(3*n) * r^(3*k) / (3*n)! in C(x,r) for n >= 0, k = 0..n, begins:
n = 0: [1];
n = 1: [2, 0];
n = 2: [40, 120, 0];
n = 3: [3680, 37440, 21600, 0];
n = 4: [880000, 20592000, 38966400, 8553600, 0];
n = 5: [435776000, 19269888000, 79491456000, 57708288000, 6329664000, 0];
n = 6: [386949376000, 28748332800000, 213892766208000, 335872728576000, 123646051584000, 7852204800000, 0];
n = 7: [560034421760000, 64544356546560000, 774705298498560000, 2169194182594560000, 1730103155573760000, 374841224017920000, 15132769090560000, 0];
n = 8: [1233482823823360000, 208114576947425280000, 3741268129758720000000, 16693947940315852800000, 23676862831649280000000, 11169319418477383680000, 1563368171330211840000, 42815371615948800000, 0];
...
in which column 0 gives the unsigned coefficients in the Dixon elliptic function cm(x,0) (cf. A104134).
RELATED SERIES.
S(x,r) = Integral C(x,r)^2 * D(x,r)^2 dx = x + (4 + 4*r^3)*x^4/4! + (160 + 800*r^3 + 160*r^6)*x^7/7! + (20800 + 292800*r^3 + 292800*r^6 + 20800*r^9)*x^10/10! + (6476800 + 191910400*r^3 + 500121600*r^6 + 191910400*r^9 + 6476800*r^12)*x^13/13! + (3946624000 + 210590336000*r^3 + 1091343616000*r^6 + 1091343616000*r^9 + 210590336000*r^12 + 3946624000*r^15)*x^16/16! + (4161608704000 + 361556726784000*r^3 + 3216369361920000*r^6 + 6333406238720000*r^9 + 3216369361920000*r^12 + 361556726784000*r^15 + 4161608704000*r^18)*x^19/19! + (6974121256960000 + 919365914368000000*r^3 + 12789764316088320000*r^6 + 42703786876467200000*r^9 + 42703786876467200000*r^12 + 12789764316088320000*r^15 + 919365914368000000*r^18 + 6974121256960000*r^21)*x^22/22! + ...
where C(x,r)^3 - S(x,r)^3 = 1.
D(x,r) = 1 + r^3 * Integral S(x,r)^2 * C(x,r)^2 dx = 1 + 2*r^3*x^3/3! + (120*r^3 + 40*r^6)*x^6/6! + (21600*r^3 + 37440*r^6 + 3680*r^9)*x^9/9! + (8553600*r^3 + 38966400*r^6 + 20592000*r^9 + 880000*r^12)*x^12/12! + (6329664000*r^3 + 57708288000*r^6 + 79491456000*r^9 + 19269888000*r^12 + 435776000*r^15)*x^15/15! + (7852204800000*r^3 + 123646051584000*r^6 + 335872728576000*r^9 + 213892766208000*r^12 + 28748332800000*r^15 + 386949376000*r^18)*x^18/18! + (15132769090560000*r^3 + 374841224017920000*r^6 + 1730103155573760000*r^9 + 2169194182594560000*r^12 + 774705298498560000*r^15 + 64544356546560000*r^18 + 560034421760000*r^21)*x^21/21! + ...
where D(x,r)^3 - r^3 * C(x,r)^3 = (1 - r^3).

Links

Crossrefs

Cf. A104134 (cm(x,0)), A357540 (S(x,r)), A357542 (D(x,r)), A178575 (row sums), A357545 (central terms).
Cf. A357801.

Programs

  • PARI
    {T(n,k) = my(S=x,C=1,D=1); for(i=0,n,
S = intformal( C^2*D^2 +O(x^(3*n+3)));
C = 1 + intformal( S^2*D^2);
D = 1 + r^3*intformal( S^2*C^2); );
(3*n)!*polcoeff( polcoeff(C,3*n,x),3*k,r)}
for(n=0,10, for(k=0,n, print1( T(n,k),", "));print(""))
  • PARI
    /* Using Series Reversion for S(x,r) (faster) */
{T(n,k) = my(S = serreverse( intformal( 1/((1 + x^3)^2*(1 + r^3*x^3)^2 +O(x^(3*n+3)) )^(1/3) )) );
(3*n)!*polcoeff( polcoeff((1 + S^3)^(1/3),3*n,x),3*k,r)}
for(n=0,10, for(k=0,n, print1( T(n,k),", "));print(""))

Formula

Generating function C(x,r) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(3*n) * r^(3*k) / (3*n)! and related functions S(x,r) and D(x,r) satisfy the following formulas.
For brevity, some formulas here will use S = S(x,r), C = C(x,r), and D = D(x,r).
(1.a) C(x,r)^3 - S(x,r)^3 = 1.
(1.b) D(x,r)^3 - r^3 * S(x,r)^3 = 1.
(1.c) D(x,r)^3 - r^3 * C(x,r)^3 = 1 - r^3.
Integral formulas.
(2.a) S(x,r) = Integral C(x,r)^2 * D(x,r)^2 dx.
(2.b) C(x,r) = 1 + Integral S(x,r)^2 * D(x,r)^2 dx.
(2.c) D(x,r) = 1 + r^3 * Integral S(x,r)^2 * C(x,r)^2 dx.
(2.d) C(x,r)^3 = 1 + 3 * Integral S(x,r)^2 * C(x,r)^2 * D(x,r)^2 dx.
Derivatives.
(3.a) d/dx S(x,r) = C(x,r)^2 * D(x,r)^2.
(3.b) d/dx C(x,r) = S(x,r)^2 * D(x,r)^2.
(3.c) d/dx D(x,r) = r^3 * S(x,r)^2 * C(x,r)^2.
Exponential formulas.
(4.a) C - S = exp( -Integral (C + S) * D^2 dx ).
(4.b) D - r*S = exp( -r * Integral (D + r*S) * C^2 dx ).
(4.c) C + S = sqrt(C^2 - S^2) * exp( Integral D^2/(C^2 - S^2) dx ).
(4.d) D + r*S = sqrt(D^2 - r^2*S^2) * exp( r * Integral C^2/(D^2 - r^2*S^2) dx ).
(5.a) C^2 - S^2 = exp( -2 * Integral S*C/(C + S) * D^2 dx ).
(5.b) D^2 - r^2*S^2 = exp( -2*r^2 * Integral S*D/(D + r*S) * C^2 dx ).
(5.c) C^2 + S^2 = exp( 2 * Integral S*C*(C + S)/(C^2 + S^2) * D^2 dx ).
(5.d) D^2 + r^2*S^2 = exp( 2*r^2 * Integral S*D*(D + r*S)/(D^2 + r^2*S^2) * C^2 dx ).
Hyperbolic functions.
(6.a) C = sqrt(C^2 - S^2) * cosh( Integral D^2/(C^2 - S^2) dx ).
(6.b) S = sqrt(C^2 - S^2) * sinh( Integral D^2/(C^2 - S^2) dx ).
(6.c) D = sqrt(D^2 - r^2*S^2) * cosh( r * Integral C^2/(D^2 - r^2*S^2) dx ).
(6.d) r*S = sqrt(D^2 - r^2*S^2) * sinh( r * Integral C^2/(D^2 - r^2*S^2) dx ).
Other formulas.
(7) S(x,r) = Series_Reversion( Integral 1/((1 + x^3)*(1 + r^3*x^3))^(2/3) dx ).
(8.a) T(n,0) = (-1)^n * A104134(n).
(8.b) Sum_{k=0..n} T(n,k) = (3*n)!/(3^n*n!) * Product_{k=1..n} (3*k - 2) = A178575(n), for n >= 0.
From Paul D. Hanna, Apr 14 2023 (Start):
Let F(x,r) = Integral 1/((1 + x^3)*(1 + r^3*x^3))^(2/3) dx, then
(9.a) S( F(x,r), r) = x,
(9.b) C( F(x,r), r) = (1 + x^3)^(1/3),
(9.c) D( F(x,r), r) = (1 + r^3*x^3)^(1/3). (End)

A357802 Coefficients T(n,k) of x^(4*n)*r^(4*k)/(4*n)! in power series D(x,r) = 1 + r^4 * Integral S(x,r)^3 * C(x,r)^3 dx such that C(x,r)^4 - S(x,r)^4 = 1 and D(x,r)^4 - r^4*S(x,r)^4 = 1, as a triangle read by rows.

Original entry on oeis.org

1, 0, 6, 0, 6048, 2268, 0, 35126784, 56282688, 7434504, 0, 679185948672, 2514356038656, 1409371197696, 95227613712, 0, 33022883487154176, 220415417637617664, 284770675495950336, 81696140755536384, 3354162536029536, 0, 3393656235362623684608, 35701050229143616880640, 83079959422282198548480, 54913653475645427527680, 9583398717725834749440, 264444869673131894208
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2022

Keywords

Comments

Equals a row reversal of triangle A357801.

Examples

			E.g.f.: D(x,r) = Sum_{n>=0} T(n,k) * x^(4*n) * r^(4*k) / (4*n)! begins:
D(x,r) = 1 + 6*r^4*x^4/4! + (6048*r^4 + 2268*r^8)*x^8/8! + (35126784*r^4 + 56282688*r^8 + 7434504*r^12)*x^12/12! + (679185948672*r^4 + 2514356038656*r^8 + 1409371197696*r^12 + 95227613712*r^16)*x^16/16! + (33022883487154176*r^4 + 220415417637617664*r^8 + 284770675495950336*r^12 + 81696140755536384*r^16 + 3354162536029536*r^20)*x^20/20! + (3393656235362623684608*r^4 + 35701050229143616880640*r^8 + 83079959422282198548480*r^12 + 54913653475645427527680*r^16 + 9583398717725834749440*r^20 + 264444869673131894208*r^24)*x^24/24! +
where D(x,r) = 1 + r^4 * Integral S(x,r)^3 * D(x,r)^3 dx.
TRIANGLE.
This triangle of coefficients T(n,k) of x^(4*n) * r^(4*k) / (4*n)! in D(x,r) for n >= 0, k = 0..n, begins:
n = 0: [1];
n = 1: [0, 6];
n = 2: [0, 6048, 2268];
n = 3: [0, 35126784, 56282688, 7434504];
n = 4: [0, 679185948672, 2514356038656, 1409371197696, 95227613712];
n = 5: [0, 33022883487154176, 220415417637617664, 284770675495950336, 81696140755536384, 3354162536029536];
n = 6: [0, 3393656235362623684608, 35701050229143616880640, 83079959422282198548480, 54913653475645427527680, 9583398717725834749440, 264444869673131894208]; ...
in which the main diagonal equals A153300.
RELATED SERIES.
S(x,r) = x + (18 + 18*r^4)*x^5/5! + (14364 + 58968*r^4 + 14364*r^8)*x^9/9! + (70203672 + 671650056*r^4 + 671650056*r^8 + 70203672*r^12)*x^13/13! + (1192064637456 + 20707300240704*r^4 + 47530354598496*r^8 + 20707300240704*r^12 + 1192064637456*r^16)*x^17/17! + (52269828456672288 + 1437626817559769760*r^4 + 5941554215913771840*r^8 + 5941554215913771840*r^12 + 1437626817559769760*r^16 + 52269828456672288*r^20)*x^21/21! + (4930307288899134335424 + 197041019249105562351744*r^4 + 1283341580573615116868160*r^8 + 2308585363008068715943680*r^12 + 1283341580573615116868160*r^16 + 197041019249105562351744*r^20 + 4930307288899134335424*r^24)*x^25/25! + ...
where D(x,r)^4 - r^4*S(x,r)^4 = 1.
C(x,r) = 1 + 6*x^4/4! + (2268 + 6048*r^4)*x^8/8! + (7434504 + 56282688*r^4 + 35126784*r^8)*x^12/12! + (95227613712 + 1409371197696*r^4 + 2514356038656*r^8 + 679185948672*r^12)*x^16/16! + (3354162536029536 + 81696140755536384*r^4 + 284770675495950336*r^8 + 220415417637617664*r^12 + 33022883487154176*r^16)*x^20/20! + (264444869673131894208 + 9583398717725834749440*r^4 + 54913653475645427527680*r^8 + 83079959422282198548480*r^12 + 35701050229143616880640*r^16 + 3393656235362623684608*r^20)*x^24/24! + ...
where D(x,r)^4 - r^4 * C(x,r)^4 = 1 - r^4.

Links

Crossrefs

Cf. A153300 (diagonal), A357805 (row sums), A357800 (S(x,r)), A357801 (C(x,r)).
Cf. A357542.

Programs

  • PARI
    {T(n, k) = my(S=x, C=1, D=1); for(i=0, n,
S = intformal( C^3*D^3 +O(x^(4*n+4)));
C = 1 + intformal( S^3*D^3);
D = 1 + r^4*intformal( S^3*C^3); );
(4*n)!*polcoeff( polcoeff(D, 4*n, x), 4*k, r)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))
  • PARI
    /* Using Series Reversion (faster) */
{T(n, k) = my(S = serreverse( intformal( 1/((1 + x^4)^3*(1 + r^4*x^4)^3 +O(x^(4*n+4)) )^(1/4) )) );
(4*n)!*polcoeff( polcoeff( (1 + r^4*S^4)^(1/4), 4*n, x), 4*k, r)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))

Formula

Generating function D(x,r) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(4*n) * r^(4*k) / (4*n)! and related functions S(x,r) and C(x,r) satisfy the following formulas.
For brevity, some formulas here will use S = S(x,r), C = C(x,r), and D = D(x,r).
(1.a) C(x,r)^4 - S(x,r)^4 = 1.
(1.b) D(x,r)^4 - r^4 * S(x,r)^4 = 1.
(1.c) D(x,r)^4 - r^4 * C(x,r)^4 = 1 - r^4.
Integral formulas.
(2.a) S(x,r) = Integral C(x,r)^3 * D(x,r)^3 dx.
(2.b) C(x,r) = 1 + Integral S(x,r)^3 * D(x,r)^3 dx.
(2.c) D(x,r) = 1 + r^4 * Integral S(x,r)^3 * C(x,r)^3 dx.
(2.d) D(x,r)^4 = 1 + r^4 * Integral 4 * S(x,r)^3 * C(x,r)^3 * D(x,r)^3 dx.
Derivatives.
(3.a) d/dx S(x,r) = C(x,r)^3 * D(x,r)^3.
(3.b) d/dx C(x,r) = S(x,r)^3 * D(x,r)^3.
(3.c) d/dx D(x,r) = r^4 * S(x,r)^3 * C(x,r)^3.
Exponential formulas.
(4.a) C + S = exp( Integral (C^2 - C*S + S^2) * D^3 dx ).
(4.b) D + r*S = exp( r * Integral (D^2 - r*D*S + r^2*S^2) * C^3 dx ).
(4.c) C - S = exp( -Integral (C^2 + C*S + S^2) * D^3 dx ).
(4.d) D - r*S = exp( -r * Integral (D^2 + r*D*S + r^2*S^2) * C^3 dx ).
(5.a) C^2 + S^2 = exp( 2 * Integral S*C * D^3 dx ).
(5.b) D^2 + r^2*S^2 = exp( 2*r^2 * Integral S*D * C^3 dx ).
(5.c) C^2 - S^2 = exp( -2 * Integral S*C * D^3 dx ).
(5.d) D^2 - r^2*S^2 = exp( -2*r^2 * Integral S*D * C^3 dx ).
Hyperbolic functions.
(6.a) C = sqrt(C^2 - S^2) * cosh( Integral (C^2 + S^2) * D^3 dx ).
(6.b) S = sqrt(C^2 - S^2) * sinh( Integral (C^2 + S^2) * D^3 dx ).
(6.c) D = sqrt(D^2 - r^2*S^2) * cosh( r * Integral (D^2 + r^2*S^2) * C^3 dx ).
(6.d) r*S = sqrt(D^2 - r^2*S^2) * sinh( r * Integral (D^2 + r^2*S^2) * C^3 dx ).
(7.a) C^2 = cosh( 2 * Integral S*C * D^3 dx ).
(7.b) S^2 = sinh( 2 * Integral S*C * D^3 dx ).
(7.c) D^2 = cosh( 2*r^2 * Integral S*D * C^3 dx ).
(7.d) r^2*S^2 = sinh( 2*r^2 * Integral S*D * C^3 dx ).
Other formulas.
(8) S(x,r) = Series_Reversion( Integral 1/((1 + x^4)*(1 + r^4*x^4))^(3/4) dx ).
(9.a) T(n,0) = T(n,n) = A153301(n).
(9.b) Sum_{k=0..n} T(n,k) = A357805(n), for n >= 0.
From Paul D. Hanna, Apr 12 2023 (Start):
Let F(x,r) = Integral 1/((1 + x^4)*(1 + r^4*x^4))^(3/4) dx, then
(10.a) S( F(x,r), r) = x,
(10.b) C( F(x,r), r) = (1 + x^4)^(1/4),
(10.c) D( F(x,r), r) = (1 + r^4*x^4)^(1/4). (End)

A357805 a(n) = coefficient of x^(4*n)/(4*n)! in power series C(x) = 1 + Integral S(x)^3 * C(x)^3 dx such that C(x)^4 - S(x)^4 = 1.

Original entry on oeis.org

1, 6, 8316, 98843976, 4698140798736, 623259279912288096, 186936162949832833285056, 110352751044119383032310847616, 116215132158682166284921510741483776, 202905498509713715271588290261091671041536, 554890365215965228675768455367962915432839248896
Offset: 0

Views

Author

Paul D. Hanna, Oct 14 2022

Keywords

Comments

Equals row sums of triangles A357801 and A357802.

Examples

			E.g.f.: C(x) = 1 + 6*x^4/4! + 8316*x^8/8! + 98843976*x^12/12! + 4698140798736*x^16/16! + 623259279912288096*x^20/20! + 186936162949832833285056*x^24/24! + 110352751044119383032310847616*x^28/28! + ...
such that
C( Integral 1/(1 + x^4)^(3/2) dx ) = (1 + x^4)^(1/4)
also
C(x)^4 - S(x)^4 = 1,
where
S(x) = x + 36*x^5/5! + 87696*x^9/9! + 1483707456*x^13/13! + 91329084354816*x^17/17! + 14862901723860427776*x^21/21! + 5279211177231308343054336*x^25/25! + ... + A357804(n)*x^(4*n+1)/(4*n+1)! + ...

Links

Crossrefs

Cf. A357804 (S(x)), A357801, A357802, A153300.

Programs

  • PARI
    /* Using Series Reversion (faster) */
{a(n) = my(S = serreverse( intformal( 1/(1 + x^4 +O(x^(4*n+4)))^(3/2) )) );
(4*n)!*polcoeff( (1 + S^4)^(1/4), 4*n)}
for(n=0, 10, print1( a(n), ", "))
  • PARI
    {a(n) = my(S=x, C=1); for(i=0, n,
S = intformal( C^6 +O(x^(4*n+4)));
C = 1 + intformal( S^3*C^3 ) );
(4*n)!*polcoeff( C, 4*n)}
for(n=0, 10, print1( a(n), ", "))

Formula

Generating function C(x) = Sum_{n>=0} a(n)*x^(4*n)/(4*n)! and related function S(x) satisfies the following formulas.
For brevity, some formulas here will use C = C(x) and S = S(x), where S(x) = (C(x)^4 - 1)^(1/4) is the e.g.f. of A357804.
(1) C(x)^4 - S(x)^4 = 1.
Integral formulas.
(2.a) S(x) = Integral C(x)^6 dx.
(2.b) C(x) = 1 + Integral S(x)^3 * C(x)^3 dx.
(2.c) S(x)^4 = Integral 4 * S(x)^3 * C(x)^6 dx.
(2.d) C(x)^4 = 1 + Integral 4 * S(x)^3 * C(x)^6 dx.
Derivatives.
(3.a) d/dx S(x) = C(x)^6.
(3.b) d/dx C(x) = S(x)^3 * C(x)^3.
Exponential formulas.
(4.a) C + S = exp( Integral (C^2 - C*S + S^2) * C^3 dx ).
(4.b) C - S = exp( -Integral (C^2 + C*S + S^2) * C^3 dx ).
(5.a) C^2 + S^2 = exp( 2 * Integral S*C^4 dx ).
(5.b) C^2 - S^2 = exp( -2 * Integral S*C^4 dx ).
Hyperbolic functions.
(6.a) C = sqrt(C^2 - S^2) * cosh( Integral (C^2 + S^2) * C^3 dx ).
(6.b) S = sqrt(C^2 - S^2) * sinh( Integral (C^2 + S^2) * C^3 dx ).
(7.a) C^2 = cosh( 2 * Integral S*C^4 dx ).
(7.b) S^2 = sinh( 2 * Integral S*C^4 dx ).
Explicit formulas.
(8.a) S(x) = Series_Reversion( Integral 1/(1 + x^4)^(3/2) dx ).
(8.b) C( Integral 1/(1 + x^4)^(3/2) dx ) = (1 + x^4)^(1/4).
Showing 1-5 of 5 results.

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