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 1-10 of 10 results.

A384581 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A143501.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 4, 0, 1, 4, 6, 10, 16, 0, 1, 5, 10, 19, 41, 92, 0, 1, 6, 15, 32, 78, 224, 616, 0, 1, 7, 21, 50, 131, 411, 1464, 4729, 0, 1, 8, 28, 74, 205, 672, 2617, 11002, 40776, 0, 1, 9, 36, 105, 306, 1031, 4170, 19251, 93234, 388057, 0
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2025

Keywords

Examples

			Square array begins:
  1,   1,    1,    1,    1,    1,    1, ...
  0,   1,    2,    3,    4,    5,    6, ...
  0,   1,    3,    6,   10,   15,   21, ...
  0,   4,   10,   19,   32,   50,   74, ...
  0,  16,   41,   78,  131,  205,  306, ...
  0,  92,  224,  411,  672, 1031, 1518, ...
  0, 616, 1464, 2617, 4170, 6245, 8997, ...

Crossrefs

Columns k=0..1 give A000007, A143501.
Cf. A381566, A384580, A384582, A384583.

Programs

  • PARI
    a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+k, j)/(3*n-3*j+k)*a(n-j, j)));

Formula

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(3*n-3*j+k,j)/(3*n-3*j+k) * A(n-j,j).

A143500 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^2).

Original entry on oeis.org

1, 1, 1, 3, 10, 46, 244, 1481, 10020, 74400, 599573, 5200284, 48223360, 475557054, 4965035754, 54672110310, 632853655686, 7678552433184, 97404631390960, 1288861146261679, 17752479062092470, 254051633672160696, 3770953046565933003, 57964955567444706668
Offset: 0

Views

Author

Paul D. Hanna, Aug 20 2008

Keywords

Examples

			G.f. A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 46*x^5 + 244*x^6 +...
A(x)^2 = 1 + 2*x + 3*x^2 + 8*x^3 + 27*x^4 + 118*x^5 + 609*x^6 +...
A(x*A(x)^2) = 1 + x + 3*x^2 + 10*x^3 + 46*x^4 + 244*x^5 +...
If G(x*A(x)^2) = x then
G(x) = x - 2*x^2 + 5*x^3 - 18*x^4 + 68*x^5 - 300*x^6 + 1283*x^7 -+...
A(G(x)) = 1 + A(x)*G(x) = (x/G(x))^(1/2) where
A(x)*G(x) = x - x^2 + 4*x^3 - 12*x^4 + 59*x^5 - 209*x^6 + 1199*x^7 -...

Links

Crossrefs

Cf. A087949, A143501.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,x*A^2));polcoeff(A,n)}
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+k, j)/(2*n-2*j+k)*a(n-j, j))); \\ Seiichi Manyama, Jun 04 2025

Formula

G.f. satisfies: G(x) = x/[1 + A(x)*G(x)]^2 = x/A(G(x))^2 where G(x*A(x)^2) = x.
From Seiichi Manyama, Jun 04 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(2*n-2*j+k,j)/(2*n-2*j+k) * a(n-j,j). (End)

A212029 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^3)^3.

Original entry on oeis.org

1, 1, 3, 21, 190, 2112, 26922, 382110, 5920788, 98862273, 1762572957, 33325846461, 664774457583, 13932829786025, 305788481726799, 7008171327166869, 167321925537782445, 4153009604547937170, 106963758805117459392, 2854029374011293902121, 78773444214057182702790
Offset: 0

Views

Author

Paul D. Hanna, Apr 27 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 190*x^4 + 2112*x^5 + 26922*x^6 +...
Related expansions:
A(x)^3 = 1 + 3*x + 12*x^2 + 82*x^3 + 732*x^4 + 7944*x^5 + 99156*x^6 +..
A(x*A(x)^3) = 1 + x + 6*x^2 + 51*x^3 + 560*x^4 + 7155*x^5 + 102495*x^6 +...
A(x*A(x)^3)^3 = 1 + 3*x + 21*x^2 + 190*x^3 + 2112*x^4 + 26922*x^5 +...

Crossrefs

Cf. A143508, A143501, A212028, A381570.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A^3, x, x*A^3)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+k, j)/(3*n-3*j+k)*a(n-j, 3*j))); \\ Seiichi Manyama, Mar 01 2025

Formula

From Seiichi Manyama, Mar 01 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(3*n-3*j+k,j)/(3*n-3*j+k) * a(n-j,3*j). (End)

A143508 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^2)^2.

Original entry on oeis.org

1, 1, 2, 9, 52, 372, 3058, 28074, 282028, 3059328, 35497672, 437499541, 5696752234, 78036803430, 1120687989348, 16823652188164, 263345788211608, 4289062071449610, 72543038644585822, 1271980596430351862, 23085579883157411532, 433071407705851089244
Offset: 0

Views

Author

Paul D. Hanna, Aug 21 2008

Keywords

Examples

			G.f. A(x) = 1 + x + 2*x^2 + 9*x^3 + 52*x^4 + 372*x^5 + 3058*x^6 +...
A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 126*x^4 + 884*x^5 + 7149*x^6 +...
A(x*A(x)^2) = 1 + x + 4*x^2 + 22*x^3 + 156*x^4 + 1285*x^5 + 11886*x^6 +...
A(x*A(x)^2)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 372*x^4 + 3058*x^5 +...
Define G(x) by G(x*A(x)^2) = x, then
G(x) = x - 2*x^2 + 3*x^3 - 12*x^4 + 17*x^5 - 198*x^6 - 345*x^7 +...
such that G(x) = x/(1 + A(x)^2*G(x))^2.

Crossrefs

Cf. A212029, A143501, A381568.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^2,x,x*A^2));polcoeff(A,n)}
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+k, j)/(2*n-2*j+k)*a(n-j, 2*j))); \\ Seiichi Manyama, Mar 01 2025

Formula

Given g.f. A(x), let G(x) be defined by G(x*A(x)^2) = x, then
(1) G(x) = x/(1 + A(x)^2*G(x))^2,
(2) A(G(x)) = 1 + A(x)^2*G(x).
From Seiichi Manyama, Mar 01 2025: (Start)
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(2*n-2*j+k,j)/(2*n-2*j+k) * a(n-j,2*j). (End)

A384574 G.f. A(x) satisfies A(x) = 1 + x * A(x*A(x)^4).

Original entry on oeis.org

1, 1, 1, 5, 23, 155, 1236, 11286, 116333, 1329433, 16630343, 225606826, 3294976854, 51496560764, 856858516809, 15112857079891, 281479726839851, 5517842789917283, 113510479973132860, 2444032094604379100, 54948814775692303024, 1287258966133883349701
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2025

Keywords

Crossrefs

Column k=1 of A384582.
Cf. A087949, A143500, A143501, A384575.
Cf. A384578.

Programs

  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n-4*j+k, j)/(4*n-4*j+k)*a(n-j, j)));

Formula

See A384582.

A384575 G.f. A(x) satisfies A(x) = 1 + x * A(x*A(x)^5).

Original entry on oeis.org

1, 1, 1, 6, 31, 236, 2166, 22722, 269889, 3567412, 51765431, 816476196, 13892821878, 253442895075, 4930644856063, 101830536332051, 2223767436058566, 51172807259226084, 1237092039069090235, 31332521053777095784, 829389782837272248191, 22894754438382163120136
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2025

Keywords

Crossrefs

Column k=1 of A384583.
Cf. A087949, A143500, A143501, A384574.
Cf. A384579.

Programs

  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n-5*j+k, j)/(5*n-5*j+k)*a(n-j, j)));

Formula

See A384583.

A212028 G.f. satisfies: A(x) = 1 + x*A(x*A(x)^3)^2.

Original entry on oeis.org

1, 1, 2, 11, 74, 635, 6296, 70268, 864106, 11546531, 165996792, 2548556963, 41546769324, 715850868468, 12986529841038, 247255748839532, 4926870211273246, 102495266879754087, 2221254395951869988, 50049980203162990978, 1170440788530570387644
Offset: 0

Views

Author

Paul D. Hanna, Apr 27 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 74*x^4 + 635*x^5 + 6296*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 26*x^3 + 174*x^4 + 1462*x^5 + 14279*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 46*x^3 + 306*x^4 + 2526*x^5 + 24311*x^6 +...
A(x*A(x)^3) = 1 + x + 5*x^2 + 32*x^3 + 273*x^4 + 2715*x^5 + 30542*x^6 + 379200*x^7 + 5117211*x^8 + 74266646*x^9 + 1150267802*x^10 +...

Crossrefs

Cf. A143501, A143508, A212029.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A^3, x, x*A^3)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

A384680 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)*A(x*A(x)^3) ).

Original entry on oeis.org

1, 1, 3, 15, 100, 805, 7442, 76750, 866818, 10586499, 138549918, 1929878820, 28459172110, 442421488758, 7225177328165, 123586748434192, 2208493015533530, 41138303109509415, 797178212982793708, 16041390159326400966, 334654194086236031816, 7227174934846895031544
Offset: 0

Views

Author

Seiichi Manyama, Jun 06 2025

Keywords

Crossrefs

Column k=1 of A384681.
Cf. A143501, A215505, A384145.

Programs

  • Mathematica
    terms = 22; A[] = 0; Do[A[x] = 1/(1-x*A[x]*A[x*A[x]^3]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jun 07 2025 *)
  • PARI
    a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-j+k, j)/(3*n-j+k)*a(n-j, j)));

Formula

See A384681.

A212030 G.f. satisfies: A(x) = 1 + x*A(x*A(x)^2)^3.

Original entry on oeis.org

1, 1, 3, 18, 142, 1350, 14607, 174626, 2263749, 31426878, 463144150, 7199095692, 117452998632, 2003613768328, 35628141598164, 658723330672311, 12636278430184303, 251042922016657782, 5156985005918404047, 109382326645948764003, 2392477607054828471286
Offset: 0

Views

Author

Paul D. Hanna, Apr 27 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 142*x^4 + 1350*x^5 + 14607*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 42*x^3 + 329*x^4 + 3092*x^5 + 33090*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 73*x^3 + 570*x^4 + 5307*x^5 + 56226*x^6 +...
A(x*A(x)^2) = 1 + x + 5*x^2 + 37*x^3 + 346*x^4 + 3745*x^5 + 45132*x^6 +...
A(x*A(x)^2)^3 = 1 + 3*x + 18*x^2 + 142*x^3 + 1350*x^4 + 14607*x^5 +...

Crossrefs

Cf. A143508, A143501, A212029.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A^3, x, x*A^2)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

A384577 G.f. A(x) satisfies A(x) = ( 1 + x * A(x*A(x))^(1/3) )^3.

Original entry on oeis.org

1, 3, 6, 19, 78, 411, 2617, 19251, 160254, 1482400, 15035622, 165545253, 1963006576, 24908182305, 336397711074, 4813816122917, 72704962269990, 1155070280657286, 19245587072017468, 335418172582313610, 6100293082529588802, 115532044092709366555, 2274095852526512246841
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2025

Keywords

Crossrefs

Column k=3 of A384581.

Programs

  • PARI
    a(n, k=3) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+k, j)/(3*n-3*j+k)*a(n-j, j)));

Formula

See A384581.
G.f.: B(x)^3, where B(x) is the g.f. of A143501.
Showing 1-10 of 10 results.

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