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.

A120970 G.f. A(x) satisfies A(x/A(x)^2) = 1 + x ; thus A(x) = 1 + Series_Reversion(x/A(x)^2).

Original entry on oeis.org

1, 1, 2, 9, 60, 504, 4946, 54430, 655362, 8496454, 117311198, 1711459903, 26228829200, 420370445830, 7021029571856, 121859518887327, 2192820745899978, 40831103986939664, 785429260324068156, 15585831041632684997, 318649154587152781210, 6704504768568697046504
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2006

Keywords

Comments

From Paul D. Hanna, Nov 16 2008: (Start)
More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1. (End)

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 60*x^4 + 504*x^5 + 4946*x^6 + ...
Related expansions.
A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 142*x^4 + 1164*x^5 + 11221*x^6 + ...
A(A(x)-1) = 1 + x + 4*x^2 + 26*x^3 + 218*x^4 + 2151*x^5 + 23854*x^6 + ...
A(A(x)-1)^2 = 1 + 2*x + 9*x^2 + 60*x^3 + 504*x^4 + 4946*x^5 + ...
x/A(x)^2 = x - 2*x^2 - x^3 - 10*x^4 - 73*x^5 - 662*x^6 - 6842*x^7 - ...
Series_Reversion(x/A(x)^2) = x + 2*x^2 + 9*x^3 + 60*x^4 + 504*x^5 + 4946*x^6 + ...
To illustrate the formula a(n) = [x^(n-1)] 2*A(x)^(2*n)/(2*n),
form a table of coefficients in A(x)^(2*n) as follows:
  A^2:  [(1), 2,   5,   22,   142,   1164,   11221,   121848, ...];
  A^4:  [ 1, (4), 14,   64,   397,   3116,   29002,   306468, ...];
  A^6:  [ 1,  6, (27), 134,   825,   6270,   56492,   580902, ...];
  A^8:  [ 1,  8,  44, (240), 1502,  11200,   98144,   983016, ...];
  A^10: [ 1, 10,  65,  390, (2520), 18672,  160115,  1565260, ...];
  A^12: [ 1, 12,  90,  592,  3987, (29676), 250730,  2399388, ...];
  A^14: [ 1, 14, 119,  854,  6027,  45458, (381010), 3582266, ...]; ...
in which the main diagonal forms the initial terms of this sequence:
[2/2*(1), 2/4*(4), 2/6*(27), 2/8*(240), 2/10*(2520), 2/12*(29676), ...].

Links

Crossrefs

Cf. A120971; variants: A120972, A120974, A120976, A030266, A067145, A107096.
Cf. related variants: A145347, A145348, A147664, A145349, A145350. - Paul D. Hanna, Nov 16 2008
Cf. A381602.

Programs

  • Mathematica
    terms = 21; A[] = 1; Do[A[x] = 1 + x*A[A[x] - 1]^2 + O[x]^j // Normal, {j, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 15 2018 *)
  • PARI
    {a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0); A[ #A]=-Vec(subst(Ser(A),x,x/Ser(A)^2))[ #A]);A[n+1]}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* This sequence is generated when k=2, m=0: A(x/A(x)^k) = 1 + x*A(x)^m */ {a(n,k=2,m=0)=local(A=sum(i=0,n-1,a(i,k,m)*x^i));if(n==0,1,polcoeff((m+k)/(m+k*n)*A^(m+k*n),n-1))} \\ Paul D. Hanna, Nov 16 2008
for(n=0,30,print1(a(n),", "))
  • PARI
    b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n+k, j)/(2*n+k)*b(n-j, 2*j)));
a(n) = if(n==0, 1, b(n-1, 2)); \\ Seiichi Manyama, Jun 04 2025

Formula

G.f. satisfies: A(x) = 1 + x*A(A(x) - 1)^2.
Let B(x) be the g.f. of A120971, then B(x) and g.f. A(x) are related by:
(a) B(x) = A(A(x)-1),
(b) B(x) = A(x*B(x)^2),
(c) A(x) = B(x/A(x)^2),
(d) A(x) = 1 + x*B(x)^2,
(e) B(x) = 1 + x*B(x)^2*B(A(x)-1)^2,
(f) A(B(x)-1) = B(A(x)-1) = B(x*B(x)^2).
a(n) = [x^(n-1)] (1/n)*A(x)^(2n) for n>=1 with a(0)=1; i.e., a(n) equals 1/n times the coefficient of x^(n-1) in A(x)^(2n) for n>=1. [Paul D. Hanna, Nov 16 2008]
From Seiichi Manyama, Jun 04 2025: (Start)
Let b(n,k) = [x^n] B(x)^k, where B(x) is the g.f. of A120971.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(2*n+k,j)/(2*n+k) * b(n-j,2*j).
a(n) = b(n-1,2) for n > 0. (End)

A147664 G.f. satisfies: A(x/A(x)) = 1 + x*A(x)^2.

Original entry on oeis.org

1, 1, 3, 15, 100, 801, 7296, 73174, 791751, 9116613, 110640310, 1405349658, 18585016509, 254855278565, 3612425924919, 52793266545585, 793851646358364, 12261570084250926, 194260753173421656, 3153098224666860712
Offset: 0

Views

Author

Paul D. Hanna, Nov 09 2008

Keywords

Comments

More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 100*x^4 + 801*x^5 + 7296*x^6 +...
A(x)^2 = 1 + 2*x + 7*x^2 + 36*x^3 + 239*x^4 + 1892*x^5 + 17019*x^6 +...
A(x/A(x)) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 239*x^5 + 1892*x^6 +...
A(x) = 1 + x*G(x)^3 where G(x) = A(x*G(x)) is the g.f. of A182953:
G(x) = 1 + x + 4*x^2 + 25*x^3 + 197*x^4 + 1797*x^5 + 18178*x^6 +...
To illustrate the formula a(n) = [x^(n-1)] 3*A(x)^(n+2)/(n+2),
form a table of coefficients in A(x)^(n+2) as follows:
A^3: [(1), 3, 12, 64, 426, 3345, 29766, 291999, ...];
A^4: [1, (4), 18, 100, 671, 5244, 46248, 449264, ...];
A^5: [1, 5, (25), 145, 985, 7686, 67305, 648085, ...];
A^6: [1, 6, 33, (200), 1380, 10782, 93922, 897402, ...];
A^7: [1, 7, 42, 266, (1869), 14658, 127246, 1207753, ...];
A^8: [1, 8, 52, 344, 2466, (19456), 168604, 1591496, ...];
A^9: [1, 9, 63, 435, 3186, 25335, (219522), 2063052, ...]; ...
in which the main diagonal forms the initial terms of this sequence:
[3/3*(1), 3/4*(4), 3/5*(25), 3/6*(200), 3/7*(1869), 3/8*(19456), ...].
ALTERNATE GENERATING METHOD.
This sequence forms column zero in the follow array.
Let A denote this sequence, and A^2 the self-convolution square of A.
Start in row zero with A^2, after prepending an initial '1', then repeat: drop the initial term and perform convolution with A and the remaining terms in a given row to obtain the next row:
[1, 1, 2, 7, 36, 239, 1892, 17019, 168746, 1807656, 20634852, 248560373, ...];
[1, 3, 12, 64, 426, 3345, 29766, 291999, 3097746, 35059659, 419160576,...];
[3, 15, 85, 571, 4443, 38952, 376090, 3930156, 43875903, 518191486, ...];
[15, 100, 701, 5494, 47883, 457451, 4724372, 52138050, 609215321, ...];
[100, 801, 6495, 56980, 542331, 5558082, 60793521, 704009142, ...];
[801, 7296, 65878, 630811, 6448878, 70124397, 806356465, 9749112990, ...];
[7296, 73174, 718577, 7386763, 80183478, 917721557, 11031656810, ...];
[73174, 791751, 8324862, 90823582, 1038161379, 12431200320, 155525155360, ...]; ...

Links

Crossrefs

Cf. A182953, A145348, A145350, A120972, A145345, A145349.

Programs

  • PARI
    {a(n)=local(F=1+x,G);for(i=0,n,G=serreverse(x/(F+x*O(x^n))^1)/x;F=1+x*G^3);polcoeff(F,n)}
  • PARI
    /* This sequence is generated when k=1, m=2: A(x/A(x)^k) = 1 + x*A(x)^m */
{a(n, k=1, m=2)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)+x*O(x^n)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* Prints terms 0..30 */
{A=[1];
for(m=1,30,
  B=Vec(Ser(A)^2);
  for(i=1,m-1, C=Vec(Ser(A)*Ser(B)); B=vector(#C-1,n,C[n+1]) );
  A=concat(A,0); A[#A]=B[1];
);
A} \\ Paul D. Hanna, Jan 10 2016

Formula

G.f.: A(x) = 1 + x*G(x)^3 where G(x) = A(x*G(x)) and A(x) = G(x/A(x)) is the g.f. of A182953.
a(n) = [x^(n-1)] 3*A(x)^(n+2)/(n+2) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 3*A(x)^(n+2)/(n+2) for n>=1 (see comment).

A145349 G.f. satisfies: A(x/A(x)^3) = 1 + x*A(x).

Original entry on oeis.org

1, 1, 4, 34, 416, 6319, 111124, 2177346, 46440184, 1061938195, 25762345804, 658072997702, 17600573291712, 490770914734054, 14219015899154068, 426904437068035200, 13252855203929697200, 424634035832800883743, 14020984603588221319072, 476460483567807053745450
Offset: 0

Views

Author

Paul D. Hanna, Nov 11 2008

Keywords

Comments

More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 416*x^4 + 6319*x^5 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 127*x^3 + 1512*x^4 + 22419*x^5 +...
A(x/A(x)^3) = 1 + x + x^2 + 4*x^3 + 34*x^4 + 416*x^5 + 6319*x^6 +...
A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^3):
G(x) = 1 + x + 7*x^2 + 82*x^3 + 1239*x^4 + 21942*x^5 + 434746*x^6 +...
G(x)^3 = 1 + 3*x + 24*x^2 + 289*x^3 + 4377*x^4 + 77097*x^5 +...
To illustrate the formula a(n) = [x^(n-1)] 4*A(x)^(3*n+1)/(3*n+1),
form a table of coefficients in A(x)^(3*n+1) as follows:
A^4: [(1), 4, 22, 188, 2217, 32516, 555972, ...];
A^7: [1, (7), 49, 441, 5131, 73248, 1220457, ...];
A^10: [1, 10, (85), 820, 9590, 134482, 2191060, ...];
A^13: [1, 13, 130, (1352), 16107, 223886, 3582072, ...];
A^16: [1, 16, 184, 2064, (25276), 351072, 5541912, ...];
A^19: [1, 19, 247, 2983, 37772, (527839), 8260174, ...];  ...
in which the main diagonal forms the initial terms of this sequence:
[4/4*(1), 4/7*(7), 4/10*(85), 4/13*(1352), 4/16*(25276), 4/19*(527839), ...].
ALTERNATE GENERATING METHOD.
This sequence forms column zero in the follow array.
Let A denote this sequence, and A^3 the self-convolution cube of A.
Start in row zero with this sequence, A, after prepending an initial '1', then repeat: drop the initial term and perform convolution with A^3 and the remaining terms in a given row to obtain the next row:
[1, 1, 1, 4, 34, 416, 6319, 111124, 2177346, 46440184, 1061938195, ...];
[1, 4, 22, 188, 2217, 32516, 555972, 10655628, 223313220, 5034249556, ...];
[4, 34, 314, 3619, 50829, 833591, 15417781, 313704516, 6900409869, ...];
[34, 416, 5071, 70714, 1131649, 20377616, 404581945, 8712077584, ...];
[416, 6319, 92167, 1472688, 26106282, 508663862, 10756749655, ...];
[6319, 111124, 1843974, 32709364, 630332565, 13142389012, 294045605744, ...];
[111124, 2177346, 39908146, 770233015, 15926231353, 352361228127, ...];
[46440184, 1061938195, 22576531219, 497127293708, 11532665230502, ...]; ...

Crossrefs

Cf. variants: A002293, A145347, A145348, A120974, A145345, A145350, A147664.

Programs

  • PARI
    {a(n)=local(A=1+x,G);for(i=0,n,G=(serreverse(x/(A+x*O(x^n))^3)/x)^(1/3);A=1+x*G^4);polcoeff(A,n)}
  • PARI
    /* This sequence is generated when k=3, m=1: A(x/A(x)^k) = 1 + x*A(x)^m */
{a(n, k=3, m=1)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)+x*O(x^n)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* Prints terms 0..30 */
{A=[1];
for(m=1,30,
  B=A;
  for(i=1,m-1, C=Vec(Ser(A)^3*Ser(B)); B=vector(#C-1,n,C[n+1]) );
  A=concat(A,0);A[#A]=B[1]
);
A} \\ Paul D. Hanna, Jan 10 2016

Formula

G.f.: A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^3) and A(x) = G(x/A(x)^3).
a(n) = [x^(n-1)] 4*A(x)^(3*n+1)/(3*n+1) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 4*A(x)^(3*n+1)/(3*n+1) for n>=1 (see comment).

A145347 G.f. satisfies: A(x/A(x)) = 1 + x*A(x)^3.

Original entry on oeis.org

1, 1, 4, 26, 220, 2203, 24836, 306104, 4047988, 56713521, 834286612, 12801754120, 203889888832, 3357619794321, 56999146850380, 995081586539016, 17830012791062632, 327376145842252333, 6151225530281186372, 118142009771446643592, 2317165307900630229384
Offset: 0

Views

Author

Paul D. Hanna, Nov 09 2008

Keywords

Comments

More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then
A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);
thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 220*x^4 + 2203*x^5 + 24836*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 103*x^3 + 876*x^4 + 8679*x^5 + 96382*x^6 +...
A(x/A(x)) = 1 + x + 3*x^2 + 15*x^3 + 103*x^4 + 876*x^5 + 8679*x^6 +...
A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)):
G(x) = 1 + x + 5*x^2 + 39*x^3 + 381*x^4 + 4284*x^5 + 53163*x^6 +...
To illustrate the formula a(n) = [x^(n-1)] 4*A(x)^(n+3)/(n+3),
form a table of coefficients in A(x)^(n+3) as follows:
A^4: [(1), 4, 22, 156, 1337, 13220, 145988, 1759876, ...];
A^5: [1, (5), 30, 220, 1905, 18836, 207100, 2481740, ...];
A^6: [1, 6, (39), 296, 2595, 25704, 281727, 3358488, ...];
A^7: [1, 7, 49, (385), 3423, 34020, 372141, 4416658, ...];
A^8: [1, 8, 60, 488, (4406), 44000, 480900, 5686480, ...];
A^9: [1, 9, 72, 606, 5562, (55881), 610872, 7202268, ...]; ...
in which the main diagonal forms the initial terms of this sequence:
[4/4*(1), 4/5*(5), 4/6*(39), 4/7*(385), 4/8*(4406), 4/9*(55881), ...].

Links

Crossrefs

Cf. A384265, A145348, A145350, A147664, A120972.

Programs

  • PARI
    {a(n) = my(F=1+x); for(i=0,n, G=serreverse(x/(F+x*O(x^n))^1)/x; F=1+x*G^4); polcoef(F,n)}
  • PARI
    /* This sequence is generated when k=1, m=3: A(x/A(x)^k) = 1 + x*A(x)^m */
{a(n, k=1, m=3) = my(A=sum(i=0, n-1, a(i, k, m)*x^i) +x*O(x^n)); if(n==0, 1, polcoef((m+k)/(m+k*n)*A^(m+k*n), n-1))}
for(n=0,20,print1(a(n),", "))

Formula

G.f.: A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)) and A(x) = G(x/A(x)).
a(n) = [x^(n-1)] 4*A(x)^(n+3)/(n+3) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 4*A(x)^(n+3)/(n+3) for n>=1 (see comment).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.