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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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.

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.

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 1-4 of 4 results.

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