A120012 -id:A120012 - cp's OEIS frontend

A127275 Expansion of (sqrt(1-4x)-x)/(1-4x).

1, 1, 2, 4, 6, -4, -100, -664, -3514, -16916, -77388, -343144, -1490148, -6376616, -26992264, -113317936, -472661434, -1961361076, -8104733884, -33374212936, -137031378124, -561253753336, -2293947547384, -9358755316816, -38121140494564, -155064370272904

Offset: 0

Paul D. Hanna, Jun 07 2006

Hankel transform is A127276.

The second self-composition of the g.f. G(x) of A120009 is G(G(x)) = (sqrt(1-4x)-x)/(1-4x) - 1.

			A(x) = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 - 4*x^5 - 100*x^6 - 664*x^7 + ...

Robert Israel, Table of n, a(n) for n = 0..1653

Cf. A120009, A120012 (3rd self-composition); A000108 (Catalan).

S:= series((sqrt(1-4*x)-x)/(1-4*x),x,31):
seq(coeff(S,x,i),i=0..30); # Robert Israel, Jan 15 2023

{a(n)=local(k=2,x=X+X^3*O(X^n));polcoeff( x*((1-k+k^2)-k^2*(k+1)*x-k*(1-(k+2)*x)*(1-sqrt(1-4*x))/2/x)/(1-k+k^2*x)^2,n,X)}

a(n) = C(2n,n) - 4^(n-1) + 0^n/4. - Paul Barry, Jan 10 2007
Conjecture: n*a(n) + 2*(-4*n+3)*a(n-1) + 8*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 26 2012
Conjecture verified using the differential equation (4*x-1)^2 * g'(x) + (8*x-2)*g(x) + 1 - 2*x = 0 satisfied by the g.f. - Robert Israel, Jan 15 2023

Definition revised by Paul Barry, Jan 10 2007

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar and Max Alekseyev

A120014 Coefficients of x^n in the n-th iteration of the g.f. of A120009, so that: a(n) = [x^n] { (x-x^2) o x/(1-nx) o (1-sqrt(1-4x))/2 } for n>=1.

Original entry on oeis.org

1, 2, 9, 60, 530, 5892, 79681, 1276760, 23729310, 502780580, 11974950746, 316917570312, 9230453871756, 293492484431720, 10117826259791025, 375952605020796720, 14980065429077943734, 637215061582781559972

Offset: 1

Paul D. Hanna, Jun 07 2006, Jun 09 2006

nonn

a(n) is divisible by n for n>=1; a(n)/n = A120016(n).

Main diagonal of table (A120013) of iterations of A120009.

			Successive iterations of F(x), the g.f. of A120009, begin:
F(x) = (1)x + x^2 + x^3 - 6x^5 - 33x^6 - 143x^7 - 572x^8 - 2210x^9 +...
F(F(x)) = x + (2)x^2 + 4x^3 + 6x^4 - 4x^5 - 100x^6 - 664x^7 +...
F(F(F(x))) = x + 3x^2 + (9)x^3 + 24x^4 + 42x^5 - 87x^6 - 1575x^7 +...
F(F(F(F(x)))) = x + 4x^2 + 16x^3 + (60)x^4 + 192x^5 + 360x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 120x^4 + (530)x^5 +1955x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 +210x^4 +1164x^5 + (5892)x^6 +...

Cf. A120016 (a(n)/n); A120009, A127275 (g.f.=F(F(x))), A120012 (g.f.=F(F(F(x)))); A000108 (Catalan); A120015, A120020, A120013.

a(n)=local(k=n,x=X+X^3*O(X^n));polcoeff( x*((1-k+k^2)-k^2*(k+1)*x-k*(1-(k+2)*x)*(1-sqrt(1-4*x))/2/x)/(1-k+k^2*x)^2,n,X)

/* Generated as the n-th self-composition of A120009: */ a(n)=local(F=((1-3*x)*sqrt(1-4*x+x^3*O(x^n)) - (1-x)*(1-4*x))/(2*x^2), G=x+x*O(x^n)); if(n<1, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, n, x)))

a(n)=n^(n-1)-sum(k=2,n-2,n^(k-1)*k*(k-1)*(n-k-1)*(2*n-k-2)!/(n-k)!)/n!

a(n) = [x^n] x*((1-n+n^2) - n^2*(n+1)*x - n*(1-(n+2)*x)*C(x) )/(1-n+n^2*x)^2, where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).

a(n) = n^(n-1) - Sum_{k=2..n-2} n^(k-1)*k*(k-1)*(n-k-1)*(2*n-k-2)!/(n-k)!/n!

A120013 Square table, read by antidiagonals, of coefficients of x^k in the n-th self-composition of the g.f. of A120009, so that: T(n,k) = [x^k] { (x-x^2) o x/(1-nx) o (1-sqrt(1-4x))/2 } for n>=1, k>=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 0, 1, 4, 9, 6, -6, 1, 5, 16, 24, -4, -33, 1, 6, 25, 60, 42, -100, -143, 1, 7, 36, 120, 192, -87, -664, -572, 1, 8, 49, 210, 530, 360, -1575, -3514, -2210, 1, 9, 64, 336, 1164, 1955, -1568, -12240, -16916, -8398, 1, 10, 81, 504, 2226, 5892, 3785, -26804, -77730, -77388, -31654

Offset: 1

Paul D. Hanna, Jun 10 2006

			Square table begins:
1, 1, 1, 0, -6, -33, -143, -572, -2210, -8398, ...
1, 2, 4, 6, -4, -100, -664, -3514, -16916, -77388, ...
1, 3, 9, 24, 42, -87, -1575, -12240, -77730, -449994, ...
1, 4, 16, 60, 192, 360, -1568, -26804, -240800, -1804456, ...
1, 5, 25, 120, 530, 1955, 3785, -28900, -508610, -5227110, ...
1, 6, 36, 210, 1164, 5892, 24552, 48258, -577380, -10814388, ...
1, 7, 49, 336, 2226, 13965, 79681, 370216, 733054, -12716578, ...
1, 8, 64, 504, 3872, 28688, 200960, 1276760, 6548320, 13015536, ...
1, 9, 81, 720, 6282, 53415, 437697, 3387636, 23729310, 133234434, ...
1, 10, 100, 990, 9660, 92460, 862120, 7743550, 65644780, 502780580,...
Successive self-compositions of F(x), the g.f. of A120009, start:
F(x) = x + x^2 + x^3 - 6x^5 - 33x^6 - 143x^7 - 572x^8 - 2210x^9 +...
F(F(x)) = x + 2x^2 + 4x^3 + 6x^4 - 4x^5 - 100x^6 - 664x^7 +...
F(F(F(x))) = x + 3x^2 + 9x^3 + 24x^4 + 42x^5 - 87x^6 - 1575x^7 +...
F(F(F(F(x)))) = x + 4x^2 + 16x^3 + 60x^4 + 192x^5 + 360x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 120x^4 + 530x^5 +1955x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 +210x^4 +1164x^5 + 5892x^6 +...

Rows: A120009, A127275, A120012; Diagonals: A120014, A120015.

{T(n,k)=sum(j=1,k,n^(j-2)*(n-j+1)*j*(2*k-j-1)!/(k-j)!/k!)}

T(n,k) = Sum_{j=1..k} n^(j-2)*(n-j+1)*j*(2*k-j-1)!/(k-j)!/k! - Paul D. Hanna and Max Alekseyev. Alternate formula: T(n,k) = n^(k-1) - Sum_{j=2..k-2} n^(j-1)*j*(j-1)*(k-j-1)*(2*k-j-2)!/(k-j)!/k!. These formulas also apply to non-integer n.

A120015 Coefficients of x^n in the (n+1)-th self-composition of the g.f. of A120009, so that: a(n) = [x^n] { (x-x^2) o x/(1-(n+1)x) o (1-sqrt(1-4x))/2 } for n>=1.

Original entry on oeis.org

1, 3, 16, 120, 1164, 13965, 200960, 3387636, 65644780, 1440018822, 35314018656, 958109355632, 28508766348664, 923461269689985, 32357613376995840, 1219728800410342556, 49225886778689380044, 2118029584754948604618

Offset: 1

Paul D. Hanna, Jun 12 2006

nonn

			Successive self-compositions of F(x), the g.f. of A120009, begin:
F(x) = x + x^2 + x^3 - 6x^5 - 33x^6 - 143x^7 - 572x^8 - 2210x^9 +...
F(F(x)) = (1)x + 2x^2 + 4x^3 + 6x^4 - 4x^5 - 100x^6 - 664x^7 +...
F(F(F(x))) = x + (3)x^2 + 9x^3 + 24x^4 + 42x^5 - 87x^6 - 1575x^7 +...
F(F(F(F(x)))) = x + 4x^2 + (16)x^3 + 60x^4 + 192x^5 + 360x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + (120)x^4 + 530x^5 +1955x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 +210x^4 + (1164)x^5 +5892x^6+...

Cf. A120014; A120009, A127275 (g.f.=F(F(x))), A120012 (g.f.=F(F(F(x)))); A120020.

{a(n)=sum(j=1, n,(n+1)^(j-2)*(n-j+2)*j*(2*n-j-1)!/(n-j)!/n!)}

a(n) = Sum_{j=1..n} (n+1)^(j-2)*(n-j+2)*j*(2*n-j-1)!/(n-j)!/n! - Paul D. Hanna and Max Alekseyev.

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A127275 Expansion of (sqrt(1-4x)-x)/(1-4x).

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A120014 Coefficients of x^n in the n-th iteration of the g.f. of A120009, so that: a(n) = [x^n] { (x-x^2) o x/(1-n*x) o (1-sqrt(1-4*x))/2 } for n>=1.

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A120013 Square table, read by antidiagonals, of coefficients of x^k in the n-th self-composition of the g.f. of A120009, so that: T(n,k) = [x^k] { (x-x^2) o x/(1-n*x) o (1-sqrt(1-4*x))/2 } for n>=1, k>=1.

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A120015 Coefficients of x^n in the (n+1)-th self-composition of the g.f. of A120009, so that: a(n) = [x^n] { (x-x^2) o x/(1-(n+1)*x) o (1-sqrt(1-4*x))/2 } for n>=1.

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PARI

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A120014 Coefficients of x^n in the n-th iteration of the g.f. of A120009, so that: a(n) = [x^n] { (x-x^2) o x/(1-nx) o (1-sqrt(1-4x))/2 } for n>=1.

A120013 Square table, read by antidiagonals, of coefficients of x^k in the n-th self-composition of the g.f. of A120009, so that: T(n,k) = [x^k] { (x-x^2) o x/(1-nx) o (1-sqrt(1-4x))/2 } for n>=1, k>=1.

A120015 Coefficients of x^n in the (n+1)-th self-composition of the g.f. of A120009, so that: a(n) = [x^n] { (x-x^2) o x/(1-(n+1)x) o (1-sqrt(1-4x))/2 } for n>=1.