A094600 -id:A094600 - cp's OEIS frontend

A094601 G.f. satisfies: A(x) = F(x*A(x)), where F(x) is the g.f. of A094600.

1, 1, 3, 12, 50, 234, 1125, 5620, 28753, 150106, 796240, 4279232, 23251672, 127518750, 704957715, 3924307492, 21978740682, 123758612644, 700204091361, 3978636187708, 22694470914700, 129904466979030, 745949776425002

Offset: 0

Paul D. Hanna, May 13 2004

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 50*x^4 + 234*x^5 + 1125*x^6 + 5620*x^7 +...
where
A(x) = Sum_{n>=1} A094600(2*n)*x^n/(n+1), and
log(A(x)) = Sum_{n>=1} A094600(2*n-1)*x^n/n,
log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 145*x^4/4 + 831*x^5/5 + 4664*x^6/6 +...

Cf. A094600, A094558.

{a(n)=local(A=1+x); for(i=1,n, A=subst(A+x*A', x, x^2*A^2)+x*A*subst(A', x, x^2*A^2)/subst(A, x, x^2*A^2) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

a(n) = A094600(2*n)/(n+1) for n>=0.
G.f.: A(x) = exp( Sum_{n>=1} A094600(2*n-1)*x^n/n ).
G.f. satisfies: A(x) = A(y) + x*A(x)*A'(y)/A(y) + x^2*A(x)^2*A'(y) where y = x^2*A(x)^2.

Entry revised by Paul D. Hanna, Apr 17 2013

A094557 a(2n) equals coefficient of x^n in A(x)^(n+1) and a(2n+1) equals coefficient of x^n in A(x)^(n+2), for n>=0.

Original entry on oeis.org

1, 1, 2, 3, 9, 14, 40, 65, 210, 339, 1080, 1764, 5775, 9448, 30992, 50931, 168849, 277920, 925240, 1525887, 5106288, 8431260, 28309440, 46796334, 157627548, 260788843, 880639004, 1458096900, 4934715105, 8175734400, 27721876064

Offset: 0

Paul D. Hanna, May 11 2004

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 9*x^4 + 14*x^5 + 40*x^6 + 65*x^7 +...
Terms are produced by main and secondary diagonals in the table of successive self-convolutions of this sequence:
  [(1), 1, 2, 3, 9, 14, 40, 65, 210, 339, 1080, ...];
  [(1),(2), 5, 10, 28, 58, 153, 320, 875, 1850, ...];
  [1, (3),(9), 22, 63, 153, 410, 978, 2607, 6222, ...];
  [1, 4, (14),(40), 121, 328, 918, 2392, 6504, 16708, ...];
  [1, 5, 20, (65),(210), 621, 1830, 5110, 14395, 39085, ...];
  [1, 6, 27, 98, (339),(1080), 3356, 9942, 29163, 83008, ...];
  [1, 7, 35, 140, 518, (1764),(5775), 18040, 55160, 163863, ...];
  [1, 8, 44, 192, 758, 2744, (9448),(30992), 98729, 305240, ...];
  [1, 9, 54, 255, 1071, 4104, 14832, (50931),(168849), 542164, ...];
  [1, 10, 65, 330, 1470, 5942, 22495, 80660, (277920),(925240), ...]; ...
from which A094558 may be formed from the main diagonal:
  [1/1, 2/2, 9/3, 40/4, 210/5, 1080/6, 5775/7, 30992/8, 168849/9, 925240/10,...].
Let G(x) be the g.f. of A094558:
G(x) = 1 + x + 3*x^2 + 10*x^3 + 42*x^4 + 180*x^5 + 825*x^6 + 3874*x^7 +...
then the coefficients of G(x)^2 generates the secondary diagonal:
[1*2/2, 3*2/3, 14*2/4, 65*2/5, 339*2/6, 1764*2/7, 9448*2/8, 50931*2/9,...]
and may be derived from the odd-indexed terms of this sequence.

Cf. A094558, A094600.

{a(n)=local(A=1+x,G);for(i=1,n,G=serreverse(x/A+x*O(x^n));A=subst(deriv(G),x,x^2)+subst(deriv(G^2/2),x,x^2)/x);polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

a(2*n) = (n+1)*A094558(n).
G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A094558.
G.f. satisfies: A(x) = F'(x^2)*(1 + F(x^2)/x) where F(x) = Series_Reversion(x/A(x)) and F(x)/x is the g.f. of A094558.

Entry revised by Paul D. Hanna, Apr 17 2013

A139796 Last term of A139687(n) with a fourth leading 1 = 1, 1, 1, 1, 2, 2, 1, 3, 5, 5 rows.

Original entry on oeis.org

1, 1, 2, 5, 9, 28, 48, 165, 275, 1001, 1638, 6188

Offset: 0

Paul Curtz, Jun 14 2008

Based on sequences identical to their p-th differences.

Cf. A094600.

Odd rows are 1; 1, 2, 2; 1, 3, 6, 9, 9; 1, 4, 10, 20, 34, 48, 48; 1, 5, 15, 35, 70, 125, 200, 275, 275; 1, 6, 21, 56, 126, 252, 461, 780, 1209, 1638, 1638; with respective sums second bisection a(2n+1)= 1, 5, 28, 165, 1001, 6188 = A025174(n+1)?

cp's OEIS Frontend

A094601 G.f. satisfies: A(x) = F(x*A(x)), where F(x) is the g.f. of A094600.

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A094557 a(2n) equals coefficient of x^n in A(x)^(n+1) and a(2n+1) equals coefficient of x^n in A(x)^(n+2), for n>=0.

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A139796 Last term of A139687(n) with a fourth leading 1 = 1, 1, 1, 1, 2, 2, 1, 3, 5, 5 rows.

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cp's OEIS Frontend

A094601 G.f. satisfies: A(x) = F(x*A(x)), where F(x) is the g.f. of A094600.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A094557 a(2*n) equals coefficient of x^n in A(x)^(n+1) and a(2*n+1) equals coefficient of x^n in A(x)^(n+2), for n>=0.

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Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A139796 Last term of A139687(n) with a fourth leading 1 = 1, 1, 1, 1, 2, 2, 1, 3, 5, 5 rows.

Views

Author

Keywords

Comments

Crossrefs

Formula

A094557 a(2n) equals coefficient of x^n in A(x)^(n+1) and a(2n+1) equals coefficient of x^n in A(x)^(n+2), for n>=0.