A127275 -id:A127275 - cp's OEIS frontend

A127276 Hankel transform of A127275.

1, 1, -2, -16, -64, -208, -608, -1664, -4352, -11008, -27136, -65536, -155648, -364544, -843776, -1933312, -4390912, -9895936, -22151168, -49283072, -109051904, -240123904, -526385152, -1149239296, -2499805184, -5419040768

Offset: 0

Paul Barry, Jan 10 2007

sign

The inverse binomial transform of this sequence yields 1, 0, -3, -8,..., which is 1 followed by the negated terms of A005563. [Paul Curtz, Dec 07 2010]

The smallest odd prime divisor of a(n) is >= 13. - Vladimir Shevelev, Feb 03 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A076616, A176126, A236999.

[2^n-n*(n+1)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Aug 11 2011

A127276:=n->2^n - n*(n + 1)*2^(n - 2); seq(A127276(n), n=0..26); # Wesley Ivan Hurt, Dec 02 2013

Table[2^n - n*(n + 1)*2^(n - 2), {n, 0, 26}] (* Wesley Ivan Hurt, Dec 02 2013 *)

Conjecture: G.f.: -(4*x-1)*(x-1) / ( (2*x-1)^3 ) and a(n) = 2^n-n*(n+1)*2^(n-2). - R. J. Mathar, Dec 11 2010

a(n) = A178987(n) - A178987(n+1). - Klaus Brockhaus, Jan 08 2011

A120012 The third self-composition of A120009; g.f.: A(x) = G(G(G(x))), where G(x) = g.f. of A120009.

Original entry on oeis.org

1, 3, 9, 24, 42, -87, -1575, -12240, -77730, -449994, -2470278, -13101228, -67823484, -344888619, -1729791975, -8581375224, -42194252106, -205940062998, -998899022898, -4819339232640, -23144643733428, -110703908388582, -527633003316726, -2506857120078336

Offset: 1

Paul D. Hanna, Jun 07 2006

sign

			A(x) = x + 3*x^2 + 9*x^3 + 24*x^4 + 42*x^5 - 87*x^6 - 1575*x^7 +...
G(x) = x + x^2 + x^3 - 6*x^5 - 33*x^6 - 143*x^7 - 572*x^8 +...
where G(x) is the g.f. of A120009 and G(G(G(x))) = A(x).

Cf. A120009, A127275 (2nd self-composition); A000108 (Catalan).

{a(n)=local(k=3,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)}

G.f.: A(x) = x*(7 - 36*x - 3*(1-5*x)*C(x) )/(2-9*x)^2 where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).

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.

cp's OEIS Frontend

A127276 Hankel transform of A127275.

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A120012 The third self-composition of A120009; g.f.: A(x) = G(G(G(x))), where G(x) = g.f. of A120009.

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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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PARI

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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.