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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.

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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.

A177409 G.f. satisfies: x = A(x) - A(x)^2 - A(A(x))^2.

Original entry on oeis.org

1, 2, 12, 108, 1208, 15536, 220832, 3390480, 55411872, 954553664, 17211258240, 323148560768, 6293245904640, 126740607526400, 2633207863038976, 56330595706808576, 1238815010325576192, 27970477203200824320
Offset: 1

Views

Author

Paul D. Hanna, Jun 01 2010

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1208*x^5 + 15536*x^6 +...
Related expansions:
A(x)^2 = x^2 + 4*x^3 + 28*x^4 + 264*x^5 + 2992*x^6 + 38496*x^7 +...
A(A(x)) = x + 4*x^2 + 32*x^3 + 344*x^4 + 4384*x^5 + 62624*x^6 +...
A(A(x))^2 = x^2 + 8*x^3 + 80*x^4 + 944*x^5 + 12544*x^6 + 182336*x^7 +...
A_{-1}(x) = x - 2*x^2 - 4*x^3 - 28*x^4 - 264*x^5 - 2992*x^6 -...
...
Illustrate A_{n}(x) = A_{n+1}(x) - A_{n+1}(x)^2 - A_{n+2}(x)^2 by the following tables of coefficients in the iterations of g.f. A(x).
Coefficients in iterations A_{n}(x), n=1..8, begin:
A_1: [1, 2, 12, 108, 1208, 15536, 220832, 3390480,...];
A_2: [1, 4, 32, 344, 4384, 62624, 973056, 16152608,...];
A_3: [1, 6, 60, 756, 10936, 173968, 2972320, 53760496,...];
A_4: [1, 8, 96, 1392, 22656, 399808, 7503616, 147999296,...];
A_5: [1, 10, 140, 2300, 41720, 811760, 16670112, 357673168,...];
A_6: [1, 12, 192, 3528, 70688, 1506656, 33688064, 783303776,...];
A_7: [1, 14, 252, 5124, 112504, 2610384, 63227808, 1586464432,...];
A_8: [1, 16, 320, 7136, 170496, 4281728, 111800832, 3014395008,...].
...
Coefficients in squared iterations A_{n}(x)^2, for n=1..8, begin:
(A_1)^2: [0, 1, 4, 28, 264, 2992, 38496, 544464, 8298080,...];
(A_2)^2: [0, 1, 8, 80, 944, 12544, 182336, 2846016, 47113792,...];
(A_3)^2: [0, 1, 12, 156, 2232, 34544, 569888, 9916112, ...];
(A_4)^2: [0, 1, 16, 256, 4320, 76800, 1429376, 27691776, ...];
(A_5)^2: [0, 1, 20, 380, 7400, 149040, 3101920, 66547024, ...];
(A_6)^2: [0, 1, 24, 528, 11664, 262912, 6064576, 143126848, ...];
(A_7)^2: [0, 1, 28, 700, 17304, 431984, 10953376, 282503760, ...];
(A_8)^2: [0, 1, 32, 896, 24512, 671744, 18586368, 520656896, ...].

Crossrefs

Cf. A292810, A177397, A276365.

Programs

  • PARI
    {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x-x^2-(A+x*O(x^n))^2)); polcoeff(A, n))}

Formula

G.f. satisfies: A_{n}(x) = A_{n+1}(x) - A_{n+1}(x)^2 - A_{n+2}(x)^2 where A_{n+1}(x) = A_{n}(A(x)) denotes iteration with A_0(x)=x.
G.f. satisfies: A(x) = A(A(x)) - A(A(x))^2 - A(A(A(x)))^2.
G.f. satisfies: x = A( x-x^2 - A(x)^2 ).
...
Given g.f. A(x), A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
. A = 1 + xA^2 + xB^2;
. B = A + xB^2 + xC^2;
. C = B + xC^2 + xD^2;
. D = C + xD^2 + xE^2; ...
. also B = A(A(x))/x, C = A(A(A(x)))/x, D = A(A(A(A(x))))/x, etc.
Showing 1-2 of 2 results.

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