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.

A384620 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of (B(x)/x)^k, where B(x) is the g.f. of A213639.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 11, 38, 0, 1, 4, 18, 86, 357, 0, 1, 5, 26, 145, 815, 3832, 0, 1, 6, 35, 216, 1389, 8758, 45189, 0, 1, 7, 45, 300, 2095, 14967, 103056, 572378, 0, 1, 8, 56, 398, 2950, 22668, 175937, 1300586, 7676653, 0, 1, 9, 68, 511, 3972, 32091, 266470, 2214012, 17368633, 107971691, 0
Offset: 0

Views

Author

Seiichi Manyama, Jun 05 2025

Keywords

Examples

			Square array begins:
  1,     1,      1,      1,      1,      1,      1, ...
  0,     1,      2,      3,      4,      5,      6, ...
  0,     5,     11,     18,     26,     35,     45, ...
  0,    38,     86,    145,    216,    300,    398, ...
  0,   357,    815,   1389,   2095,   2950,   3972, ...
  0,  3832,   8758,  14967,  22668,  32091,  43488, ...
  0, 45189, 103056, 175937, 266470, 377620, 512705, ...

Crossrefs

Columns k=0..1 give A000007, A213639(n+1).
Cf. A379599, A384619, A384621, A384623.

Programs

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

Formula

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

A213591 G.f. A(x) satisfies A( x - A(x)^2 ) = x.

Original entry on oeis.org

1, 1, 4, 24, 178, 1512, 14152, 142705, 1528212, 17211564, 202460400, 2474708496, 31310415376, 408815254832, 5495451727376, 75907303147652, 1075685334980240, 15618612118252960, 232102241507321384, 3526880759915999016, 54755450619399484512, 867928449982022915984
Offset: 1

Views

Author

Paul D. Hanna, Jun 14 2012

Keywords

Comments

Unsigned version of A139702.
Self-convolution is A276370.
Row sums of triangle A277295.

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
where A(x) = x + A(A(x))^2:
A(A(x)) = x + 2*x^2 + 10*x^3 + 69*x^4 + 568*x^5 + 5250*x^6 + 52792*x^7 +...
A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
The g.f. satisfies the series:
A(x) = x + A(x)^2 + d/dx A(x)^4/2! + d^2/dx^2 A(x)^6/3! + d^3/dx^3 A(x)^8/4! +...
Logarithmic series:
log(A(x)/x) = A(x)^2/x + [d/dx A(x)^4/x]/2! + [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! +...
Also, A(x) = x*G(A(x)^2/x) where G(x) = x/A(x/G(x)^2) is the g.f. of A212411:
G(x) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 235*x^5 + 1792*x^6 + 15261*x^7 +...
Also, A(x)^2 = x*F(A(x)) where F(x) is the g.f. of A213628:
F(x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 616*x^6 + 5072*x^7 + 46013*x^8 +...

Links

Crossrefs

Cf. A220379, A139702, A212411, A138740, A213628, A213639.
Cf. A088714, A088717, A091713, A120971, A140094, A140095.
Cf. A143426, A087949, A143435, A182969.
Cf. A275765, A276360, A276361, A276362, A276363, A276370.
Cf. A277295 (triangle).

Programs

  • Mathematica
    terms = 22; A[] = 0; Do[A[x] = x + A[A[x]]^2 + O[x]^(terms+1) // Normal, terms+1]; CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Jan 09 2018 *)
  • PARI
    {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^2+x*O(x^n))); polcoeff(A, n))}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1,21,print1(a(n),", "))
  • PARI
    b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*b(n-j, 2*j)));
a(n) = b(n-1, 1); \\ Seiichi Manyama, Jun 05 2025

Formula

G.f. satisfies:
(1) A(x) = x + A(A(x))^2.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n)/x / n! ).
(4) A(x) = x*G(A(x)^2/x) where G(x) = 1 + x*G(1-1/G(x))^2 is the g.f. of A212411.
(5) A(x)^2 = x*F(A(x)) where F(x) = 1 - (x^2/F(x))/F(x^2/F(x)) is the g.f. of A213628.
(6) x = A(A( x-x^2 - A(x)^2 )). - Paul D. Hanna, Jul 01 2012
(7) A(x) is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = x + B^2;
B = A + C^2;
C = B + D^2;
D = C + E^2; ...
where B = A(A(x)), C = A(A(A(x))), D = A(A(A(A(x)))), etc.
...
a(n) = Sum_{k=0..n-1} A277295(n,k).
From Seiichi Manyama, Jun 05 2025: (Start)
Let b(n,k) = [x^n] (A(x)/x)^k.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(n+j+k,j)/(n+j+k) * b(n-j,2*j).
a(n) = b(n-1,1). (End)

A376176 G.f. A(x) satisfies x = A( x - A(x)^4/x^2 ).

Original entry on oeis.org

1, 1, 6, 55, 622, 8015, 113164, 1711898, 27357970, 457507917, 7952476482, 142972019125, 2648639456048, 50415218306637, 983728646223556, 19641163430509505, 400671660024507294, 8340743906266061866, 176998642509849677206, 3825680705425292568049, 84159282700462688412042
Offset: 1

Views

Author

Paul D. Hanna, Sep 21 2024

Keywords

Examples

			G.f.: A(x) = x + x^2 + 6*x^3 + 55*x^4 + 622*x^5 + 8015*x^6 + 113164*x^7 + 1711898*x^8 + 27357970*x^9 + 457507917*x^10 + ...
where x = A( x - A(x)^4/x^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 13*x^4 + 122*x^5 + 1390*x^6 + 17934*x^7 + 252847*x^8 + 3814724*x^9 + ...
A(x)^3 = x^3 + 3*x^4 + 21*x^5 + 202*x^6 + 2322*x^7 + 30030*x^8 + 423111*x^9 + 6369930*x^10 + ...
where A(x)^3 = x*A(x)^2 + A(A(x))^4.
A(x)^4 = x^4 + 4*x^5 + 30*x^6 + 296*x^7 + 3437*x^8 + 44600*x^9 + 628454*x^10 + 9446280*x^11 + ...
A(A(x))^4 = x^4 + 8*x^5 + 80*x^6 + 932*x^7 + 12096*x^8 + 170264*x^9 + 2555206*x^10 + 40413484*x^11 + ...
where A(x) = x + A(A(x))^4 / A(x)^2.
A(A(x)) = x + 2*x^2 + 14*x^3 + 141*x^4 + 1712*x^5 + 23392*x^6 + 347444*x^7 + 5498681*x^8 + 91552406*x^9 + ...
A(A(x))^2/A(x) = x + 3*x^2 + 23*x^3 + 242*x^4 + 3017*x^5 + 41965*x^6 + 631381*x^7 + 10089533*x^8 + 169256922*x^9 + ...

Links

Crossrefs

Cf. A213591, A213639.

Programs

  • PARI
    {a(n) = my(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^4/x^2 +x*O(x^n))); polcoeff(A, n))}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(4*m)/x^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(4*m)/x^(2*m+1))/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*b(n-j, 4*j)));
a(n) = b(n-1, 1); \\ Seiichi Manyama, Jun 05 2025

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = A( x - A(x)^4/x^2 ).
(2) A(x)^3 = x*A(x)^2 + A(A(x))^4.
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(4*n)/x^(2*n) / n!.
(4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(4*n)/x^(2*n+1) / n! ).
From Seiichi Manyama, Jun 05 2025: (Start)
Let b(n,k) = [x^n] (A(x)/x)^k.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(n+j+k,j)/(n+j+k) * b(n-j,4*j).
a(n) = b(n-1,1). (End)

A384622 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) * A(x*A(x))^5 ).

Original entry on oeis.org

1, 1, 7, 75, 989, 14822, 242833, 4253818, 78573475, 1516124048, 30358711661, 627789264431, 13357722853019, 291611321803145, 6517101781199460, 148833150175812360, 3468184751644757228, 82363850033966966043, 1991430772785525516280, 48980124394583747435367
Offset: 0

Views

Author

Seiichi Manyama, Jun 05 2025

Keywords

Crossrefs

Column k=1 of A384623.
Cf. A088714, A213591, A213639, A376176.

Programs

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

Formula

See A384623.
Showing 1-4 of 4 results.

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