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.

Showing 1-4 of 4 results.

A357545 Central terms of triangle A357541: a(n) = A357541(2*n,n).

Original entry on oeis.org

1, 120, 38966400, 335872728576000, 23676862831649280000000, 7884265450248813494550528000000, 9001018126678397460727568113336320000000, 28542885018291526761600709316931461578752000000000, 216619327660243309425808505300579182909935738421248000000000
Offset: 0

Views

Author

Paul D. Hanna, Oct 10 2022

Keywords

Comments

Also equals central terms of triangle A357542: a(n) = A357542(2*n,n).

Crossrefs

Cf. A357541, A357542.

Programs

  • PARI
    {A357541(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, 15, print1( A357541(2*n, n), ", "))

A357540 Coefficients T(n,k) of x^(3*n+1)*r^(3*k)/(3*n+1)! in power series S(x,r) = Integral C(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 symmetric triangle read by rows.

Original entry on oeis.org

1, 4, 4, 160, 800, 160, 20800, 292800, 292800, 20800, 6476800, 191910400, 500121600, 191910400, 6476800, 3946624000, 210590336000, 1091343616000, 1091343616000, 210590336000, 3946624000, 4161608704000, 361556726784000, 3216369361920000, 6333406238720000, 3216369361920000, 361556726784000, 4161608704000, 6974121256960000, 919365914368000000, 12789764316088320000, 42703786876467200000
Offset: 0

Views

Author

Paul D. Hanna, Oct 09 2022

Keywords

Comments

Related to Dixon elliptic function sm(x,0) (cf. A104133).

Examples

			E.g.f.: S(x,r) = Sum_{n>=0} T(n,k) * x^(3*n+1) * r^(3*k) / (3*n+1)! begins:
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! + ...
This table of coefficients T(n,k) of x^(3*n+1) * r^(3*k) / (3*n+1)! in S(x,r) for n >= 0, k = 0..n, begins:
n = 0: [1];
n = 1: [4, 4];
n = 2: [160, 800, 160];
n = 3: [20800, 292800, 292800, 20800];
n = 4: [6476800, 191910400, 500121600, 191910400, 6476800];
n = 5: [3946624000, 210590336000, 1091343616000, 1091343616000, 210590336000, 3946624000];
n = 6: [4161608704000, 361556726784000, 3216369361920000, 6333406238720000, 3216369361920000, 361556726784000, 4161608704000];
n = 7: [6974121256960000, 919365914368000000, 12789764316088320000, 42703786876467200000, 42703786876467200000, 12789764316088320000, 919365914368000000, 6974121256960000];
n = 8: [17455222222028800000, 3313522085749145600000, 67574136526308966400000, 348431220691544883200000, 588750579021316096000000, 348431220691544883200000, 67574136526308966400000, 3313522085749145600000, 17455222222028800000];
...
in which both column 0 and the main diagoal give the unsigned coefficients in the Dixon elliptic function sm(x,0) (cf. A104133).
RELATED SERIES.
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! + ...
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 * S(x,r)^3 = 1.

Links

Crossrefs

Cf. A104133 (sm(x,0)), A357541 (C(x,r)), A357542 (D(x,r)), A357543 (row sums), A357544 (central terms).
Cf. A357800.

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+1)!*polcoeff( polcoeff(S,3*n+1,x),3*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^3)^2*(1 + r^3*x^3)^2 +O(x^(3*n+3)) )^(1/3) )) );
(3*n+1)!*polcoeff( polcoeff(S,3*n+1,x),3*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^(3*n+1) * r^(3*k) / (3*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)^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) S(x,r)^3 = Integral 3 * 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 + x^3)^2 * (1 + r^3*x^3)^2 )^(-1/3) dx ).
(8.a) T(n,0) = T(n,n) = (-1)^n * A104133(n).
(8.b) Sum_{k=0..n} T(n,k) = (3*n+1)!/(3^n*n!) * Product_{k=1..n} (3*k - 2) = A357543(n), for n >= 0.

A357801 Coefficients T(n,k) of x^(4*n)*r^(4*k)/(4*n)! in power series C(x,r) = 1 + Integral S(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 triangle read by rows.

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Oct 14 2022

Keywords

Comments

Equals a row reversal of triangle A357802.

Examples

			E.g.f.: C(x,r) = Sum_{n>=0} T(n,k) * x^(4*n) * r^(4*k) / (4*n)! begins:
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 S(x,r) = 1 + 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 C(x,r) for n >= 0, k = 0..n, begins:
n = 0: [1];
n = 1: [6, 0];
n = 2: [2268, 6048, 0];
n = 3: [7434504, 56282688, 35126784, 0];
n = 4: [95227613712, 1409371197696, 2514356038656, 679185948672, 0];
n = 5: [3354162536029536, 81696140755536384, 284770675495950336, 220415417637617664, 33022883487154176, 0];
n = 6: [264444869673131894208, 9583398717725834749440, 54913653475645427527680, 83079959422282198548480, 35701050229143616880640, 3393656235362623684608, 0]; ...
in which column 0 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 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 * C(x,r)^4 = 1 - r^4.

Links

Crossrefs

Cf. A153300 (column 0), A357805 (row sums), A357800 (S(x,r)), A357802 (D(x,r)).
Cf. A357541.

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(C, 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 + 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 C(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 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) C(x,r)^4 = 1 + 4 * Integral S(x,r)^3 * C(x,r)^3 * D(x,r)^3 dx.
(2.e) D(x,r)^4 = 1 + 4*r^4 * Integral 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)

A357542 Coefficients T(n,k) of x^(3*n)*r^(3*k)/(3*n)! in power series D(x,r) = 1 + r^3 * 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, 0, 2, 0, 120, 40, 0, 21600, 37440, 3680, 0, 8553600, 38966400, 20592000, 880000, 0, 6329664000, 57708288000, 79491456000, 19269888000, 435776000, 0, 7852204800000, 123646051584000, 335872728576000, 213892766208000, 28748332800000, 386949376000, 0, 15132769090560000, 374841224017920000, 1730103155573760000, 2169194182594560000, 774705298498560000, 64544356546560000, 560034421760000
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 A357541 which describes the related function C(x,r).

Examples

			E.g.f.: D(x,r) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(3*n) * r^(3*k) / (3*n)! begins:
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! + ...
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: [0, 2];
n = 2: [0, 120, 40];
n = 3: [0, 21600, 37440, 3680];
n = 4: [0, 8553600, 38966400, 20592000, 880000];
n = 5: [0, 6329664000, 57708288000, 79491456000, 19269888000, 435776000];
n = 6: [0, 7852204800000, 123646051584000, 335872728576000, 213892766208000, 28748332800000, 386949376000];
n = 7: [0, 15132769090560000, 374841224017920000, 1730103155573760000, 2169194182594560000, 774705298498560000, 64544356546560000, 560034421760000];
n = 8: [0, 42815371615948800000, 1563368171330211840000, 11169319418477383680000, 23676862831649280000000, 16693947940315852800000, 3741268129758720000000, 208114576947425280000, 1233482823823360000];
...
in which the main diagonal 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 D(x,r)^3 - r^3 * S(x,r)^3 = 1.
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! + ...
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)), A357541 (C(x,r)), A178575 (row sums), A357545 (central terms).
Cf. A357802.

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(D,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 + r^3*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 D(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 C(x,r) satisfy the following relations.
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) D(x,r)^3 = 1 + r^3 * Integral 3 * 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 + x^3)^2 * (1 + r^3*x^3)^2 )^(-1/3) dx ).
(8.a) T(n,n) = (-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.
Showing 1-4 of 4 results.

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