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 23 results. Next

A384866 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 A213093.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 4, 0, 1, 4, 6, 10, 13, 0, 1, 5, 10, 19, 35, 62, 0, 1, 6, 15, 32, 69, 158, 297, 0, 1, 7, 21, 50, 119, 303, 760, 1523, 0, 1, 8, 28, 74, 190, 516, 1453, 3868, 8091, 0, 1, 9, 36, 105, 288, 821, 2462, 7359, 20487, 43243, 0
Offset: 0

Views

Author

Seiichi Manyama, Jun 11 2025

Keywords

Comments

A(42,1) = -16825305705383790675462237694.

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,  13,  35,   69,  119,  190,  288, ...
  0,  62, 158,  303,  516,  821, 1248, ...
  0, 297, 760, 1453, 2462, 3900, 5913, ...

Crossrefs

Columns k=0..1 give A000007, A213093.

Programs

  • PARI
    b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-4*n+5*j+k-1, j-1)*b(n-j, j)/j));
a(n, k) = b(n, -k);

Formula

Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-4*n+5*j+k-1,j-1) * b(n-j,j)/j. Then A(n,k) = b(n,-k).

A213091 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^2).

Original entry on oeis.org

1, 1, 1, 2, 4, 11, 31, 98, 317, 1070, 3685, 12928, 45924, 164552, 593398, 2148288, 7796846, 28328601, 102948125, 373955584, 1357252616, 4921292287, 17828236695, 64546901169, 233660589210, 846258569786, 3068523234989, 11147449003438, 40600425590874, 148330067463010
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare definition of g.f. to:
(1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
(2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 is the g.f. of the ternary tree numbers (A001764).
The first negative term is a(80). - Georg Fischer, Feb 16 2019

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 11*x^5 + 31*x^6 + 98*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 3*x^2 + 6*x^3 + 13*x^4 + 34*x^5 + 96*x^6 + 296*x^7 +...
A(-x*A(x)^2) = 1 - x - x^2 - x^3 - 4*x^4 - 10*x^5 - 34*x^6 - 107*x^7 -...

Links

Crossrefs

Cf. A000108, A001764, A213092, A213093, A213094, A213095, A213096.

Programs

  • Mathematica
    nmax = 29; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x/A[(-x) A[x]^2]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A,x,-x*subst(A^2,x,x+x*O(x^n))) );polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A213096 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^6)^3.

Original entry on oeis.org

1, 1, 3, 15, 82, 549, 3957, 31423, 262905, 2309655, 20954053, 195219912, 1855139472, 17913275558, 175183126884, 1731034436637, 17250012004590, 173095950538881, 1746875271992760, 17712083908168204, 180276161806773003, 1840572737915529150, 18838475627464850819
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare definition of g.f. to:
(1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
(2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 is the g.f. of the ternary tree numbers (A001764).
The first negative term is a(193). - Georg Fischer, Feb 16 2019

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 15*x^3 + 82*x^4 + 549*x^5 + 3957*x^6 +...
Related expansions:
A(x)^6 = 1 + 6*x + 33*x^2 + 200*x^3 + 1272*x^4 + 8730*x^5 + 63628*x^6 +...
A(-x*A(x)^6)^3 = 1 - 3*x - 6*x^2 - 19*x^3 - 156*x^4 - 1065*x^5 - 9165*x^6 -...

Links

Crossrefs

Cf. A000108, A001764, A213091, A213092, A213093, A213094, A213095.

Programs

  • Mathematica
    m = 22; A[] = 1; Do[A[x] = 1 + x/A[-x A[x]^6 + O[x]^m]^3 // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 05 2019 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A^3,x,-x*subst(A^6,x,x+x*O(x^n))) );polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A213094 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^4)^2.

Original entry on oeis.org

1, 1, 2, 7, 26, 123, 622, 3490, 20468, 125643, 792606, 5118050, 33612998, 223770400, 1505528080, 10213807498, 69746716716, 478693572991, 3298184837434, 22790090901504, 157803590073220, 1094189186549354, 7593267782966708, 52713912426435111, 365948276764762712
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare definition of g.f. to:
(1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
(2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 is the g.f. of the ternary tree numbers (A001764).
The first negative term is a(136). - Georg Fischer, Feb 16 2019

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 26*x^4 + 123*x^5 + 622*x^6 + 3490*x^7 +...
Related expansions:
A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 237*x^4 + 1112*x^5 + 5614*x^6 +...
A(-x*A(x)^4)^2 = 1 - 2*x - 3*x^2 - 6*x^3 - 38*x^4 - 180*x^5 - 1095*x^6 -...

Links

Crossrefs

Cf. A000108, A001764, A213091, A213092, A213093, A213095, A213096.

Programs

  • Mathematica
    nmax = 24; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x/A[-x A[x]^4]^2) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A^2,x,-x*subst(A^4,x,x+x*O(x^n))) );polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A213095 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^5)^2.

Original entry on oeis.org

1, 1, 2, 9, 40, 242, 1528, 10664, 76956, 575245, 4395910, 34131621, 268146598, 2122399923, 16884293154, 134689290877, 1075641369024, 8588548510081, 68496446989330, 545303352881863, 4331918361300882, 34337864000400360, 271657823631727330, 2146133623039711577
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare definition of g.f. to:
(1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
(2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 is the g.f. of the ternary tree numbers (A001764).
The first negative term is a(85). - Georg Fischer, Feb 16 2019

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 26*x^4 + 123*x^5 + 622*x^6 + 3490*x^7 +...
Related expansions:
A(x)^5 = 1 + 5*x + 20*x^2 + 95*x^3 + 485*x^4 + 2801*x^5 + 17560*x^6 +...
A(-x*A(x)^5)^2 = 1 - 2*x - 5*x^2 - 12*x^3 - 93*x^4 - 550*x^5 - 3981*x^6 -...

Links

Crossrefs

Cf. A000108, A001764, A213091, A213092, A213093, A213094, A213096.

Programs

  • Mathematica
    m = 23; A[] = 1; Do[A[x] = 1 + x/A[-x A[x]^5 + O[x]^m]^2 // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 05 2019 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A^2,x,-x*subst(A^5,x,x+x*O(x^n))) );polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A213092 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^3).

Original entry on oeis.org

1, 1, 1, 3, 8, 31, 120, 511, 2234, 9988, 45497, 208435, 959496, 4414091, 20252947, 92586100, 421351615, 1910531192, 8647504950, 39194735661, 178643040883, 822295086652, 3836023988259, 18167435295220, 87268076036356, 423657019406289, 2067868784722846
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare definition of g.f. to:
(1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
(2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 is the g.f. of the ternary tree numbers (A001764).
The first negative term is a(54) = -4736158402689145255029229896601957. - Georg Fischer, Feb 16 2019

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 8*x^4 + 31*x^5 + 120*x^6 + 511*x^7 +...
Related expansions:
A(x)^3 = 1 + 3*x + 6*x^2 + 16*x^3 + 48*x^4 + 171*x^5 + 664*x^6 + 2760*x^7 +...
A(-x*A(x)^3) = 1 - x - 2*x^2 - 3*x^3 - 14*x^4 - 50*x^5 - 213*x^6 - 915*x^7 -...

Links

Crossrefs

Cf. A000108, A001764, A213091, A213093, A213094, A213095, A213096.

Programs

  • Mathematica
    nmax = 26; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x/A[-x A[x]^3]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A,x,-x*subst(A^3,x,x+x*O(x^n))) );polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A213101 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^8)^4.

Original entry on oeis.org

1, 1, 4, 26, 188, 1627, 15172, 154904, 1670836, 18951217, 222682164, 2693625128, 33309537808, 419311915217, 5354144473084, 69169422070152, 902237854706616, 11863641066687085, 157052133090437332, 2090929291636792824, 27971914781646817864, 375725009230868446500
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare definition of g.f. to:
(1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
(2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 (A000108).
(3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 (A001764).
(4) E(x) = 1 + x/E(-x*E(x)^7)^4 when E(x) = 1 + x*E(x)^4 (A002293).
(5) F(x) = 1 + x/F(-x*F(x)^9)^5 when F(x) = 1 + x*F(x)^5 (A002294).
The first negative term is a(249). - Georg Fischer, Feb 16 2019

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 188*x^4 + 1627*x^5 + 15172*x^6 +...
Related expansions:
A(x)^8 = 1 + 8*x + 60*x^2 + 488*x^3 + 4150*x^4 + 37600*x^5 + 358788*x^6 +...
A(-x*A(x)^8)^4 = 1 - 4*x - 10*x^2 - 44*x^3 - 439*x^4 - 3884*x^5 - 42724*x^6 -...

Links

Crossrefs

Cf. A000108, A001764, A002293, A002294, A213091, A213092, A213093, A213094, A213095, A213096, A213098, A213099, A213100, A213102, A213103, A213104, A213105.

Programs

  • Mathematica
    m = 22; A[] = 1; Do[A[x] = 1 + x/A[-x A[x]^8]^4 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 06 2019 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A^4,x,-x*subst(A^8,x,x+x*O(x^n))) );polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A213102 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^9)^4.

Original entry on oeis.org

1, 1, 4, 30, 240, 2433, 26388, 315726, 3958452, 51863952, 698988716, 9637772716, 135161761860, 1920878419569, 27583547221596, 399310273694328, 5817100622299116, 85152975761167179, 1251046169511714720, 18428780031111768466, 271964652432415737596
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare definition of g.f. to:
(1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
(2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 (A000108).
(3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 (A001764).
(4) E(x) = 1 + x/E(-x*E(x)^7)^4 when E(x) = 1 + x*E(x)^4 (A002293).
(5) F(x) = 1 + x/F(-x*F(x)^9)^5 when F(x) = 1 + x*F(x)^5 (A002294).
The first negative term is a(142). - Georg Fischer, Feb 16 2019

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 30*x^3 + 240*x^4 + 2433*x^5 + 26388*x^6 +...
Related expansions:
A(x)^9 = 1 + 9*x + 72*x^2 + 642*x^3 + 6030*x^4 + 61551*x^5 + 670344*x^6 +...
A(-x*A(x)^9)^4 = 1 - 4*x - 14*x^2 - 64*x^3 - 797*x^4 - 8188*x^5 - 104090*x^6 -...

Links

Crossrefs

Cf. A000108, A001764, A002293, A002294, A213091, A213092, A213093, A213094, A213095, A213096, A213098, A213099, A213100, A213101, A213103, A213104, A213105.

Programs

  • Mathematica
    m = 21; A[] = 1; Do[A[x] = 1 + x/A[-x A[x]^9]^4 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 06 2019 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A^4,x,-x*subst(A^9,x,x+x*O(x^n))) );polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A213103 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^12)^4.

Original entry on oeis.org

1, 1, 4, 42, 420, 5779, 83104, 1306684, 21283504, 356648125, 6100611232, 105634585546, 1845124077000, 32368064972555, 568794055227200, 9991239094888864, 175142529040285920, 3060545399532144497, 53279047286232892928, 923884653765128839312, 15965368274611453269820
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare definition of g.f. to:
(1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
(2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 (A000108).
(3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 (A001764).
(4) E(x) = 1 + x/E(-x*E(x)^7)^4 when E(x) = 1 + x*E(x)^4 (A002293).
(5) F(x) = 1 + x/F(-x*F(x)^9)^5 when F(x) = 1 + x*F(x)^5 (A002294).
The first negative term is a(76). - Georg Fischer, Feb 16 2019

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 42*x^3 + 420*x^4 + 5779*x^5 + 83104*x^6 +...
Related expansions:
A(x)^12 = 1 + 12*x + 114*x^2 + 1252*x^3 + 14775*x^4 + 193956*x^5 +...
A(-x*A(x)^12)^4 = 1 - 4*x - 26*x^2 - 148*x^3 - 2415*x^4 - 33192*x^5 -...

Links

Crossrefs

Cf. A000108, A001764, A002293, A002294, A213091, A213092, A213093, A213094, A213095, A213096, A213098, A213099, A213100, A213101, A213102, A213104, A213105.

Programs

  • Mathematica
    m = 21; A[] = 1; Do[A[x] = 1 + x/A[-x A[x]^12]^4 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 06 2019 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A^4,x,-x*subst(A^12,x,x+x*O(x^n))) );polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A213104 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^10)^5.

Original entry on oeis.org

1, 1, 5, 40, 360, 3820, 43651, 543240, 7146185, 98885725, 1420274645, 21037156031, 319127602075, 4935547265370, 77525696636995, 1233356748777015, 19829269320322346, 321631227310756885, 5255920261950786655, 86436636022328320125, 1429253483704685851315
Offset: 0

Views

Author

Paul D. Hanna, Jun 05 2012

Keywords

Comments

Compare definition of g.f. to:
(1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
(2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 (A000108).
(3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 (A001764).
(4) E(x) = 1 + x/E(-x*E(x)^7)^4 when E(x) = 1 + x*E(x)^4 (A002293).
(5) F(x) = 1 + x/F(-x*F(x)^9)^5 when F(x) = 1 + x*F(x)^5 (A002294).
(6) G(x) = 1 + x/G(-x*G(x)^11)^6 when G(x) = 1 + x*G(x)^6 (A002295).
The first negative term is a(306). - Georg Fischer, Feb 16 2019

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 40*x^3 + 360*x^4 + 3820*x^5 + 43651*x^6 +...
Related expansions:
A(x)^10 = 1 + 10*x + 95*x^2 + 970*x^3 + 10335*x^4 + 116452*x^5 +...
A(-x*A(x)^10)^5 = 1 - 5*x - 15*x^2 - 85*x^3 - 995*x^4 - 10776*x^5 -...

Links

Crossrefs

Cf. A000108, A001764, A002293, A002294, A002295, A213091, A213092, A213093, A213094, A213095, A213096, A213098, A213099, A213100, A213101, A213102, A213103, A213105.

Programs

  • Mathematica
    m = 21; A[] = 1; Do[A[x] = 1 + x/A[-x A[x]^10]^5 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 06 2019 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A^5,x,-x*subst(A^10,x,x+x*O(x^n))) );polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
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