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.

A277295 G.f. A(x,y) satisfies: A( x - y*A(x,y)^2, y) = x + (1-y)*A(x,y)^2, where the coefficients T(n,k) of x^n*y^k form a triangle read by rows n>=1, for k=0..n-1.

Original entry on oeis.org

1, 1, 0, 2, 2, 0, 5, 14, 5, 0, 14, 74, 76, 14, 0, 42, 352, 698, 378, 42, 0, 132, 1588, 5088, 5404, 1808, 132, 0, 429, 6946, 32461, 56410, 37546, 8484, 429, 0, 1430, 29786, 189940, 486550, 535410, 244220, 39446, 1430, 0, 4862, 126008, 1046190, 3690410, 6036632, 4597402, 1522466, 182732, 4862, 0, 16796, 527900, 5511440, 25518020, 57890956, 66031704, 36873036, 9227504, 846248, 16796, 0, 58786, 2195580, 28061890, 164565240, 493085566, 784844330, 661152388, 281873618, 54885974, 3926338, 58786, 0
Offset: 1

Views

Author

Paul D. Hanna, Oct 11 2016

Keywords

Comments

More generally, we have the following related identity.
Given functions F and G with F(0)=0, F'(0)=1, G(0)=0, G'(0)=0,
if F(x - y*G(x)) = x + (1-y)*G(x), then
(1) F(x) = x + G( y*F(x) + (1-y)*x ),
(2) y*F(x) + (1-y)*x = Series_Reversion(x - y*G(x)),
(3) F(x) = x + G(x + y*G(x + y*G(x + y*G(x +...)))),
(4) F(x) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) G(x)^n / n!.
The g.f. of this sequence A(x,y) equals F(x) in the above when G(x) = F(x)^2.

Examples

			G.f.: A(x,y)  = x + x^2 + (2*y + 2)*x^3 + (5*y^2 + 14*y + 5)*x^4 + (14*y^3 + 76*y^2 + 74*y + 14)*x^5 + (42*y^4 + 378*y^3 + 698*y^2 + 352*y + 42)*x^6 + (132*y^5 + 1808*y^4 + 5404*y^3 + 5088*y^2 + 1588*y + 132)*x^7 + (429*y^6 + 8484*y^5 + 37546*y^4 + 56410*y^3 + 32461*y^2 + 6946*y + 429)*x^8 + (1430*y^7 + 39446*y^6 + 244220*y^5 + 535410*y^4 + 486550*y^3 + 189940*y^2 + 29786*y + 1430)*x^9 + (4862*y^8 + 182732*y^7 + 1522466*y^6 + 4597402*y^5 + 6036632*y^4 + 3690410*y^3 + 1046190*y^2 + 126008*y + 4862)*x^10 +...
such that
A( x - y*A(x,y)^2, y)  =  x + (1-y)*A(x,y)^2.
Also,
A(x,y) = x + A( y*A(x,y) + (1-y)*x, y)^2.
...
This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) begins:
1;
1, 0;
2, 2, 0;
5, 14, 5, 0;
14, 74, 76, 14, 0;
42, 352, 698, 378, 42, 0;
132, 1588, 5088, 5404, 1808, 132, 0;
429, 6946, 32461, 56410, 37546, 8484, 429, 0;
1430, 29786, 189940, 486550, 535410, 244220, 39446, 1430, 0;
4862, 126008, 1046190, 3690410, 6036632, 4597402, 1522466, 182732, 4862, 0;
16796, 527900, 5511440, 25518020, 57890956, 66031704, 36873036, 9227504, 846248, 16796, 0;
58786, 2195580, 28061890, 164565240, 493085566, 784844330, 661152388, 281873618, 54885974, 3926338, 58786, 0; ...
RELATED SEQUENCES.
Given T(n,k) is the coefficient of x^n*y^k in g.f. A(x,y),
if b(n) = Sum_{k=0..n-1} T(n,k) * p^k * q^(n-k-1)
then B(x) = Sum_{n>=1} b(n)*x^n satisfies
(1) B(x - p*B(x)^2) = x + (q-p)*B(x)^2
(2) B(x)  =  x + B( p*B(x) + (q-p)*x )^2.
Examples:
A213591(n) = sum(k=0,n-1, T(n,k) )
A275765(n) = sum(k=0,n-1, T(n,k) * 2^(n-k) )
A276360(n) = sum(k=0,n-1, T(n,k) * 3^(n-k-1) )
A276361(n) = sum(k=0,n-1, T(n,k) * 2^k * 3^(n-k-1) )
A276362(n) = sum(k=0,n-1, T(n,k) * 4^(n-k-1) )
A276363(n) = sum(k=0,n-1, T(n,k) * 3^k * 4^(n-k-1) )
A276365(n) = sum(k=0,n-1, T(n,k) * 2^k )
A277300(n) = sum(k=0,n-1, T(n,k) * 5^(n-k-1) )
A277301(n) = sum(k=0,n-1, T(n,k) * 2^k * 5^(n-k-1) )
A277302(n) = sum(k=0,n-1, T(n,k) * 3^k * 5^(n-k-1) )
A277303(n) = sum(k=0,n-1, T(n,k) * 4^k * 5^(n-k-1) )
A277304(n) = sum(k=0,n-1, T(n,k) * 6^(n-k-1) )
A277305(n) = sum(k=0,n-1, T(n,k) * 5^k * 6^(n-k-1) )
A277306(n) = sum(k=0,n-1, T(n,k) * (-1)^k )
A277307(n) = sum(k=0,n-1, T(n,k) * 3^k )
A277308(n) = sum(k=0,n-1, T(n,k) * 3^k * 2^(n-k-1) )
A277309(n) = sum(k=0,n-1, T(n,k) * 5^k * 2^(n-k-1) )
A277310(n) = sum(k=0,n-1, T(n,k) * 4^k )
A277311(n) = sum(k=0,n-1, T(n,k) * 5^k )
...

Links

Crossrefs

Cf. A000108 (column 0), A138156 (column 1), A277296 (column 2), A277297 (diagonal), A277298 (central terms T(2*n-1,n-1)), A277299 (central terms T(2*n,n-1)).
Cf. A213591 (row sums), A275765, A276360, A276361, A276362, A276363, A276365.
Cf. A277300, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309, A277310, A277311.

Programs

  • Mathematica
    c[n_] := c[n] = Module[{A}, A[x_] = x; Do[A[x_] = x + A[y A[x] + (1-y) x + x O[x]^j]^2, {j, n}] // Normal; SeriesCoefficient[A[x], {x, 0, n}] // Expand];
T[n_, k_] := SeriesCoefficient[c[n], {y, 0, k}];
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {T(n,k) = my(A=x); for(i=1, n, A = x + subst(A^2, x, y*A + (1-y)*x +x*O(x^n)) ); polcoeff(polcoeff(A,n,x),k,y)}
for(n=1, 12, for(k=0,n-1, print1(T(n,k), ", "));print(""))

Formula

G.f. A(x,y) also satisfies:
(1) A(x,y) = x + A( y*A(x,y) + (1-y)*x, y)^2.
(2) y*A(x,y) + (1-y)*x = Series_Reversion( x - y*A(x,y)^2 ).
(3) y*x + (1-y)*B(x,y) = Series_Reversion( x + (1-y)*A(x,y)^2 ), where B( A(x,y), y) = x.
(4) A(x,y) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) A(x,y)^(2*n) / n!.
In formulas 2 and 3, the series reversion is taken with respect to variable x.
T(n+1,0) = T(n+1,n-1) = binomial(2*n,n)/(n+1) = A000108(n) for n>=1.
T(n+1,1) = 4^n - (3*n+1)*binomial(2*n,n)/(n+1) = A138156(n-1) for n>=1.

A277297 Diagonal of triangle A277295; a(n) = A277295(n+2,n).

Original entry on oeis.org

2, 14, 76, 378, 1808, 8484, 39446, 182732, 846248, 3926338, 18276832, 85436832, 401313288, 1894847846, 8994916236, 42931067910, 206002472336, 993664391720, 4817140123620, 23464959242520, 114820471485840, 564248307992850, 2783898831160512, 13786409317705920, 68509284536815640, 341538385855855064, 1707729163812908528
Offset: 1

Views

Author

Paul D. Hanna, Oct 11 2016

Keywords

Comments

G.f. G(x,y) of triangle A277295 satisfies:
(1) G(x,y) = x + G( y*G(x,y) + (1-y)*x, y)^2.
(2) G( x - y*G(x,y)^2, y) = x + (1-y)*G(x,y)^2.

Crossrefs

Cf. A277295, A277296, A277298, A277299, A291822.

Programs

  • PARI
    {A277295(n,k) = my(A=x); for(i=1, n, A = x + subst(A^2, x, y*A + (1-y)*x +x*O(x^n)) ); polcoeff(polcoeff(A,n,x),k,y)}
for(n=1,30,print1(A277295(n+2,n-1),", "))

Formula

a(n) = 2 * A291822(n) for n >= 1. - Paul D. Hanna, Jul 24 2023

A277296 Column 2 of triangle A277295; a(n) = A277295(n+2,2).

Original entry on oeis.org

0, 5, 76, 698, 5088, 32461, 189940, 1046190, 5511440, 28061890, 139089880, 674600980, 3213923072, 15084364221, 69904629316, 320443984742, 1455102018480, 6553035703798, 29296998868904, 130133568013164, 574698326054848, 2524813186117010, 11040163152179976, 48069234287414668, 208481908639356448, 900995206054983396, 3881108155492099760
Offset: 1

Views

Author

Paul D. Hanna, Oct 11 2016

Keywords

Comments

G.f. G(x,y) of triangle A277295 satisfies:
(1) G(x,y) = x + G( y*G(x,y) + (1-y)*x, y)^2.
(2) G( x - y*G(x,y)^2, y) = x + (1-y)*G(x,y)^2.

Crossrefs

Cf. A277295, A277297, A277298, A277299.

Programs

  • PARI
    {A277295(n,k) = my(A=x); for(i=1, n, A = x + subst(A^2, x, y*A + (1-y)*x +x*O(x^n)) ); polcoeff(polcoeff(A,n,x),k,y)}
for(n=1,30,print1(A277295(n+2,2),", "))

A277298 Central terms of triangle A277295 in odd-indexed rows; a(n) = A277295(2*n-1,n-1).

Original entry on oeis.org

1, 2, 76, 5404, 535410, 66031704, 9583782716, 1584902128648, 292586323812088, 59462699504146980, 13171171610678351360, 3155802303880338506184, 813045290972961285049576, 224143228528852050484555760, 65850557807967931843625040120, 20543425063284611418233827507248, 6784261706348459523065200262509390
Offset: 1

Views

Author

Paul D. Hanna, Oct 11 2016

Keywords

Comments

G.f. G(x,y) of triangle A277295 satisfies:
(1) G(x,y) = x + G( y*G(x,y) + (1-y)*x, y)^2.
(2) G( x - y*G(x,y)^2, y) = x + (1-y)*G(x,y)^2.

Crossrefs

Cf. A277295, A277296, A277297, A277299.

Programs

  • PARI
    {A277295(n,k) = my(A=x); for(i=1, n, A = x + subst(A^2, x, y*A + (1-y)*x +x*O(x^n)) ); polcoeff(polcoeff(A,n,x),k,y)}
for(n=1,20,print1(A277295(2*n-1,n-1),", "))
Showing 1-4 of 4 results.

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

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