A179201 -id:A179201 - cp's OEIS frontend

A179200 E.g.f. equals the real part of the i-th iteration of (x + x^2), where i=sqrt(-1).

0, 1, 0, -6, 60, -600, 5880, -38640, -624960, 45077760, -1773129600, 58531809600, -1657462435200, 33703750080000, 171919752076800, -76383384045696000, 6034124486347776000, -348318907415331840000, 15862493882862941184000

Offset: 0

Paul D. Hanna, Jul 02 2010

sign

Let H(x) equal the i-th iteration of (x + x^2), then
. the inverse of H(x) equals the conjugate of H(x);
. H(x+x^2) = H(x) + H(x)^2;
. H(x) = F(x) + i*G(x) where G(x) = e.g.f. of A179201 and F(x) = e.g.f. of this sequence, where H(F(x) - i*G(x)) = x;
. coefficients of H(x) form the first column of triangular matrix A030528 raised to the i-th power, where A030528(n,k) = C(k,n-k).

			E.g.f: F(x) = x - 6*x^3/3! + 60*x^4/4! - 600*x^5/5! + 5880*x^6/6! +...
The e.g.f. of A179201, G(x), begins:
G(x) = 2*x^2/2! - 6*x^3/3! + 12*x^4/4! + 200*x^5/5! - 6240*x^6/6! + 139440*x^7/7! - 2869440*x^8/8! +...
The i-th iteration of (x + x^2) = H(x) = F(x) + i*G(x), begins:
H(x) = x + i*x^2 - (1 + i)*x^3 + (5 + i)*x^4/2 - (15 - 5*i)*x^5/3 + (49 - 52*i)*x^6/6 - (23 - 83*i)*x^7/3 - (93 + 427*i)*x^8/6 + (15652 + 18537*i)*x^9/126 - (61567 + 24585*i)*x^10/126 + (369519 - 42094*i)*x^11/252 - (1743963 - 1222750*i)*x^12/504 + ...
where H(F(x) - i*G(x)) = x.

Cf. A179201, A030528.

{a(n)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(c,r-c))),L=sum(k=1,#M,-(M^0-M)^k/k),N=sum(k=0,#L,(I*L)^k/k!));if(n<1,0,real(n!*N[n,1]))}

E.g.f.: F(x) satisfies:
. F(x) = (G(x+x^2)/G(x) - 1)/2
. G(x) = sqrt( F(x) + F(x)^2 - F(x+x^2) )
where G(x) is the e.g.f. of A179201.

A179198 Matrix log of triangle A030528, where A030528(n,k) = C(k,n-k).

Original entry on oeis.org

0, 1, 0, -2, 2, 0, 9, -4, 3, 0, -64, 18, -6, 4, 0, 620, -128, 27, -8, 5, 0, -7536, 1240, -192, 36, -10, 6, 0, 109032, -15072, 1860, -256, 45, -12, 7, 0, -1809984, 218064, -22608, 2480, -320, 54, -14, 8, 0, 33562944, -3619968, 327096, -30144, 3100, -384, 63, -16

Offset: 0

Paul D. Hanna, Jul 09 2010

sign

			Triangle L begins:
0;
1,0;
-2,2,0;
9,-4,3,0;
-64,18,-6,4,0;
620,-128,27,-8,5,0;
-7536,1240,-192,36,-10,6,0;
109032,-15072,1860,-256,45,-12,7,0;
-1809984,218064,-22608,2480,-320,54,-14,8,0;
33562944,-3619968,327096,-30144,3100,-384,63,-16,9,0;
-681799680,67125888,-5429952,436128,-37680,3720,-448,72,-18,10,0;
14980204800,-1363599360,100688832,-7239936,545160,-45216,4340,-512,81,-20,11,0; ...
where column_k = (k+1)*column_0: L(n,k) = (k+1)*L(n-k,0).

Cf. A179199 (column 0), A179200, A179201, A030528.

{L(n,k)=local(A030528=matrix(n+1,n+1,r,c,if(r>=c,binomial(c,r-c))),LOG,ID=A030528^0); LOG=sum(m=1,n+1,-(ID-A030528)^m/m);(n-k)!*LOG[n+1,k+1]}

L(n,k) = (k+1)*L(n-k,0).

E.g.f. of column 0 satisfies: G(x) = (1+x)/(1+2*x)*G(x+x^2); more formulas given in A179199.

cp's OEIS Frontend

A179200 E.g.f. equals the real part of the i-th iteration of (x + x^2), where i=sqrt(-1).

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