A102054 -id:A102054 - cp's OEIS frontend

A102055 Column 1 of A102054, the matrix inverse of A060083 (Euler polynomials).

1, 2, 1, 4, -13, 142, -1931, 36296, -893273, 27927346, -1081725559, 50861556172, -2854289486309, 188475382997654, -14467150771771043, 1277417937676246672, -128570745743431055281, 14632875988040732946106, -1869882665740777942166543, 266593648798424693540514836

Offset: 0

Paul D. Hanna, Dec 28 2004

sign

1-a(n+1) equals the n-th partial sum of the Genocchi numbers (A001469).

Cf. A001469, A060083, A102054, A102056.

{a(n)=local(M=matrix(n+2,n+2));M[1,1]=1;if(n>0,M[2,1]=1;M[2,2]=1); for(r=3,n+2, for(c=1,r,M[r,c]=if(c==1,M[r-1,1], if(c==r,1,M[r,c]=M[r-1,c]-((matrix(r-1,r-1,i,j,M[i,j]))^-1)[r-1,c-1])))); return(if(n==0,1,M[n+2,2]))}

a(n) = 1 - Sum_{k=1, n} A001469(k) for n>0, with a(0)=1.

This sequence's twin numbers are given in A133135. - Paul Curtz, Aug 07 2008

A102056 Column 2 of A102054, the matrix inverse of A060083 (Euler polynomials).

Original entry on oeis.org

1, 3, -2, 26, -229, 3181, -59700, 1469380, -45938639, 1779367231, -83663906454, 4695118012798, -310029578255977, 23797509154723361, -2101268283286737704, 211490399673452191176, -24070116208693613274035, 3075833697885980827017235, -438528974913900528707713514

Offset: 0

Paul D. Hanna, Dec 28 2004

sign

Cf. A060083, A102054, A102055.

PARI

A060083 Coefficients of even-indexed Euler polynomials (rising powers without zeros).

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -3, 5, -3, 1, 17, -28, 14, -4, 1, -155, 255, -126, 30, -5, 1, 2073, -3410, 1683, -396, 55, -6, 1, -38227, 62881, -31031, 7293, -1001, 91, -7, 1, 929569, -1529080, 754572, -177320, 24310, -2184, 140, -8, 1, -28820619

Offset: 0

Wolfdieter Lang, Mar 29 2001

E(2*n,1/2)*(-4)^n = A000364(n) (signless Euler numbers without zeros).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 809.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. Cigler, q-Fibonacci polynomials and q-Genocchi numbers, arXiv:0908.1219
H.-C. Herbig, D. Herden, C. Seaton, On compositions with x^2/(1-x), arXiv preprint arXiv:1404.1022, 2014

A060082 (falling powers).
Matrix inverse is A102054. Column 0 is A001469 (Genocchi numbers).
Cf. A102054, A001469.

t[n_, k_] := Binomial[2*n, 2*k]*2*(n - k)*EulerE[2*(n - k) - 1, 0]/(2*k + 1); t[n_, n_] = 1; Table[t[n, k], {n, 0, 9}, {k, 0, n }] // Flatten (* Jean-François Alcover, Jul 03 2013 *)

{T(n,k)=local(X=x+x*O(x^(2*n)),Y=y+y*O(y^(2*k+1))); (2*n)!*polcoeff(polcoeff((cosh(X*Y)*(Y-1)+ exp(X*Y)/(exp(X)+1)+exp(-X*Y)/(exp(-X)+1))/Y,2*n,x),2*k,y)} (Hanna)

E(2*n, x)= sum(a(n, m)*x^(2*m+1), m=0..n-1) + x^(2*n), n >= 1; E(0, x)=1.
T(n, k) = A102054(n, k+1) - A102054(n+1, k+1), where A102054 is matrix inverse. E.g.f.: A(x^2, y^2) = [cosh(xy)*(y-1) + exp(xy)/(exp(x)+1) + exp(-xy)/(exp(-x)+1)]/y. - Paul D. Hanna, Dec 28 2004
T(n,k) = 1/(2*k+1)*binomial(2*n,2*k)*A001469(n-k) for 0 <= k <= n-1.
Let F(n,x) = Sum_{k=0..n-1} binomial(n-k-1,k)*x^k be a Fibonacci polynomial (see A011973 for coefficients). Then F(2*n,x) = -Sum_{k=0..n-1} T(n,k)*F(2*k+1,x). For example, F(8,x) = -17*F(1,x) + 28*F(3,x) - 14*F(5,x) + 4*F(7,x). See Cigler, Corollary 1.3. - Peter Bala, Mar 14 2012

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A102055 Column 1 of A102054, the matrix inverse of A060083 (Euler polynomials).

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A102056 Column 2 of A102054, the matrix inverse of A060083 (Euler polynomials).

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A060083 Coefficients of even-indexed Euler polynomials (rising powers without zeros).

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