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 1-5 of 5 results.

A128325 Rectangular table, read by antidiagonals, where the g.f.s of row n, R(x,n), satisfy: R(x,n+1) = R(G(x),n) for n>=0 and x*R(x,0) = G(x) = x + x*G(G(x)) is the g.f. of A030266.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 12, 23, 1, 1, 5, 20, 57, 104, 1, 1, 6, 30, 114, 305, 531, 1, 1, 7, 42, 200, 712, 1787, 2982, 1, 1, 8, 56, 321, 1435, 4772, 11269, 18109, 1, 1, 9, 72, 483, 2608, 10900, 33896, 75629, 117545, 1, 1, 10, 90, 692, 4389, 22219, 86799
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Comments

Row n equals 1 + (n+2)-th self-composition of the g.f. G(x) of A030266: R(x,0) = 1 + G(G(x)); R(x,1) = 1 + G(G(G(x))); R(x,2) = 1 + G(G(G(G(x)))); etc.

Examples

			Consider the infinite system of simultaneous equations:
  A = 1 + x*A*B;
  B = 1 + x*A*B*C;
  C = 1 + x*A*B*C*D;
  D = 1 + x*A*B*C*D*E;
  E = 1 + x*A*B*C*D*E*F; ...
The unique solution to the variables are:
  A = R(x,0), B = R(x,1), C = R(x,2), D = R(x,3), E = R(x,4), etc.,
where R(x,n) denotes the g.f. of row n of this table and satisfies:
  R(x,1) = R(x*A,0); R(x,2) = R(x*A,1); R(x,3) = R(x*A,2); etc.
The row g.f.s are also related by:
  R(x,0) = 1 + x/(1 - x*R(x,1) - x*R(x,2));
  R(x,1) = 1 + x/(1 - x*R(x,1) - x*R(x,2) - x*R(x,3));
  R(x,2) = 1 + x/(1 - x*R(x,1) - x*R(x,2) - x*R(x,3) - x*R(x,4)); etc.
The initial rows of this table begin:
  R(x,0): [1, 1,  2,   6,   23,   104,    531,    2982,    18109, ...];
  R(x,1): [1, 1,  3,  12,   57,   305,   1787,   11269,    75629, ...];
  R(x,2): [1, 1,  4,  20,  114,   712,   4772,   33896,   253102, ...];
  R(x,3): [1, 1,  5,  30,  200,  1435,  10900,   86799,   720074, ...];
  R(x,4): [1, 1,  6,  42,  321,  2608,  22219,  196910,  1805899, ...];
  R(x,5): [1, 1,  7,  56,  483,  4389,  41531,  406441,  4095749, ...];
  R(x,6): [1, 1,  8,  72,  692,  6960,  72512,  777888,  8559852, ...];
  R(x,7): [1, 1,  9,  90,  954, 10527, 119832, 1399755, 16720998, ...];
  R(x,8): [1, 1, 10, 110, 1275, 15320, 189275, 2392998, 30865353, ...];
  R(x,9): [1, 1, 11, 132, 1661, 21593, 287859, 3918189, 54301621, ...];
  R(x,10):[1, 1, 12, 156, 2118, 29624, 423956, 6183400, 91673594, ...]; ...

Crossrefs

Cf. A030266 (row 0), A128326 (row 1), A128327 (row 2), A128328 (row 3), A128329 (main diagonal); A128330 (variant).

Programs

  • PARI
    {T(n,k)=local(A=vector(n+k+3,m,1+x+x*O(x^(n+k)))); for(i=1,n+k+3,for(j=1,n+k+1,N=n+k+2-j; A[N]=1+x/(1-x*sum(m=2,N+2,A[m]+x*O(x^(n+k))))));Vec(A[n+1])[k+1]}

Formula

Let R(x,n) denote the g.f. of row n of this table, then
R(x,n) = 1 + x*Product_{k=0..n+1} R(x,k),
R(x,n) = 1 + x/[1 - x*Sum_{k=1..n+2} R(x,k) ].

A128326 G.f.: A(x) = 1 + G(G(G(x))), where G(x) = x + x*G(G(x)) is the g.f. of A030266.

Original entry on oeis.org

1, 1, 3, 12, 57, 305, 1787, 11269, 75629, 535960, 3987913, 31021693, 251445581, 2117993712, 18499513147, 167246537937, 1562556275281, 15066167302802, 149737897716757, 1532313152898208, 16129331500727047
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Comments

Equals row 1 of table A128325.

Crossrefs

Cf. A030266; A128325 (table), A128327 (row 2), A128328 (row 3), A128329 (main diagonal).

Programs

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

Formula

G.f. satisfies: A(x) = x/(1 - A( x/(1 - A(x)) )) when offset is taken to be 1. - Paul D. Hanna, Dec 20 2014

A128327 Row 2 of table A128325.

Original entry on oeis.org

1, 1, 4, 20, 114, 712, 4772, 33896, 253102, 1975610, 16054568, 135413280, 1182664740, 10675334958, 99437919664, 954581258020, 9433732288486, 95883201181772, 1001411775057322, 10738668800872594, 118151145186400408
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Crossrefs

Cf. A030266; A128325 (table), A128326 (row 1), A128328 (row 3), A128329 (main diagonal).

Programs

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

Formula

G.f.: A(x) = 1 + G(G(G(G(x)))) = B(G(x)), where B(x) is the g.f. of A128326 and G(x) = x + x*G(G(x)) is the g.f. of A030266.

A128328 Row 3 of table A128325.

Original entry on oeis.org

1, 1, 5, 30, 200, 1435, 10900, 86799, 720074, 6196295, 55135043, 506125404, 4784680169, 46516469860, 464550190798, 4761343733469, 50044839978614, 539051253692777, 5946806890025709, 67156408547628636, 775935817487472046
Offset: 0

Views

Author

Paul D. Hanna, Mar 11 2007

Keywords

Crossrefs

Cf. A030266; A128325 (table), A128326 (row 1), A128327 (row 2), A128329 (main diagonal).

Programs

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

Formula

G.f.: A(x) = 1 + G(G(G(G(G(x))))) = B(G(x)), where B(x) is the g.f. of A128327 and G(x) = x + x*G(G(x)) is the g.f. of A030266.

A141141 The main diagonal in the table of coefficients of iterations of G(x), where G(x) = x + x*G(G(x)) = g.f. of A030266.

Original entry on oeis.org

1, 2, 12, 114, 1435, 22219, 406441, 8559852, 203792337, 5409449156, 158350300141, 5066765087000, 175908765569628, 6585443884172129, 264428161094825151, 11335716352419699208, 516717363793695685925, 24955728581736822645816
Offset: 1

Views

Author

Paul D. Hanna, Jun 10 2008

Keywords

Examples

			a(n) = the n-th coefficient of the n-th iteration of G(x):
[x] G(x) = 1, [x^2] G(G(x)) = 2, [x^3] G(G(G(x))) = 12, etc.
The initial iterations (n=1..7) of G(x) are:
n=1: x + x^2 + 2*x^3 + 6*x^4 + 23*x^5 + 104*x^6 + 531*x^7 +...
n=2: x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 531*x^6 + 2982*x^7 +...
n=3: x + 3*x^2 + 12*x^3 + 57*x^4 + 305*x^5 + 1787*x^6 + 11269*x^7 +...
n=4: x + 4*x^2 + 20*x^3 + 114*x^4 + 712*x^5 + 4772*x^6 + 33896*x^7 +...
n=5: x + 5*x^2 + 30*x^3 + 200*x^4 + 1435*x^5 + 10900*x^6 + 86799*x^7+...
n=6: x + 6*x^2 + 42*x^3 + 321*x^4 + 2608*x^5 + 22219*x^6 + 196910*x^7+...
n=7: x + 7*x^2 + 56*x^3 + 483*x^4 + 4389*x^5 + 41531*x^6 + 406441*x^7+...
Notice the main diagonal of the table formed from these coefficients.

Crossrefs

Cf. A030266, A128325, A128329.

Programs

  • PARI
    {a(n)=local(A=x,B); if(n<1, 0, for(i=1, n, A=serreverse(x/(1+A +x*O(x^n)))); B=x;for(i=1,n,B=subst(A,x,B+x*O(x^n)));polcoeff(B,n))}
Showing 1-5 of 5 results.

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