A060083 -id:A060083 - 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

A102054 Triangular matrix, read by rows, where T(n,k) = T(n-1,k) - [T^-1](n-1,k-1); also equals the matrix inverse of A060083 (Euler polynomials).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 4, -2, 4, 1, 1, -13, 26, -10, 5, 1, 1, 142, -229, 116, -25, 6, 1, 1, -1931, 3181, -1567, 371, -49, 7, 1, 1, 36296, -59700, 29464, -6922, 952, -84, 8, 1, 1, -893273, 1469380, -725108, 170398, -23358, 2100, -132, 9, 1, 1, 27927346, -45938639, 22669816, -5327198, 730252, -65526, 4152

Offset: 0

Paul D. Hanna, Dec 28 2004

Column 1 forms A102055. Column 2 forms A102056.

			T(5,3) = -10 = T(4,3) - A060083(4,2) = 4 - 14.
T(6,2) = -229 = T(5,2) - A060083(5,1) = 26 - 255.
Rows begin:
[1],
[1,1],
[1,2,1],
[1,1,3,1],
[1,4,-2,4,1],
[1,-13,26,-10,5,1],
[1,142,-229,116,-25,6,1],
[1,-1931,3181,-1567,371,-49,7,1],
[1,36296,-59700,29464,-6922,952,-84,8,1],...
The matrix inverse is equal to A060083:
[1],
[ -1,1],
[1,-2,1],
[ -3,5,-3,1],
[17,-28,14,-4,1],
[ -155,255,-126,30,-5,1],...

Cf. A060083, A102055, A102056.

{T(n,k)=local(M=matrix(n+1,n+1));M[1,1]=1;if(n>0,M[2,1]=1;M[2,2]=1); for(r=3,n+1, 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(M[n+1,k+1])}

T(n, k) = T(n-1, k) - A060083(n-1, k-1), for n>0, with T(0, 0)=1.

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

A004172 Triangle of coefficients of Euler polynomials E_2n(x) (exponents in increasing order).

Original entry on oeis.org

1, 0, -1, 1, 0, 1, 0, -2, 1, 0, -3, 0, 5, 0, -3, 1, 0, 17, 0, -28, 0, 14, 0, -4, 1, 0, -155, 0, 255, 0, -126, 0, 30, 0, -5, 1, 0, 2073, 0, -3410, 0, 1683, 0, -396, 0, 55, 0, -6, 1, 0, -38227, 0, 62881, 0, -31031, 0, 7293, 0, -1001, 0, 91, 0, -7, 1, 0, 929569, 0

Offset: 0

N. J. A. Sloane

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.

T. D. Noe, Rows n = 0..50 of triangle, flattened
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].
Eric Weisstein's World of Mathematics, Euler Polynomial.

Cf. A060083.

Flatten[Table[CoefficientList[EulerE[2n, x], x], {n, 0, 8}]] (* Jean-François Alcover, Jul 21 2011 *)

A060082 Coefficients of even-indexed Euler polynomials (falling powers without zeros).

Original entry on oeis.org

1, 1, -1, 1, -2, 1, 1, -3, 5, -3, 1, -4, 14, -28, 17, 1, -5, 30, -126, 255, -155, 1, -6, 55, -396, 1683, -3410, 2073, 1, -7, 91, -1001, 7293, -31031, 62881, -38227, 1, -8, 140, -2184, 24310, -177320, 754572, -1529080, 929569, 1, -9, 204, -4284, 67626, -753610, 5497596, -23394924, 47408019

Offset: 0

Wolfdieter Lang, Mar 29 2001

E(2n,x) = x^(2n) + Sum_{k=1..n} a(n,k)*x^(2n-2k+1).

			E(0,x) = 1.
E(2,x) = x^2 - x.
E(4,x) = x^4 - 2*x^3 + x.
E(6,x) = x^6 - 3*x^5 + 5*x^3 - 3*x.
E(8,x) = x^8 - 4*x^7 + 14*x^5 - 28*x^3 + 17*x.
E(10,x) = x^10 - 5*x^9 + 30*x^7 - 126*x^5 + 255*x^3 - 155*x.

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.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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].
Z.-W. Sun, Introduction to Bernoulli and Euler polynomials

E(2n, 1/2)*(-4)^n = A000364(n) (signless Euler numbers without zeros).
-E(2n, -1/2)*(-4)^n/3 = A076552(n), -E(2n, 1/3)*(-9)^n/2 = A002114(n).
Cf. A060083 (rising powers), A060096-7 (Euler polynomials), A004172 (with zeros).
Columns (left edge) include A000330, A053132. Columns (right edge) include A001469.

Table[ CoefficientList[ EulerE[2*n, x], x] // Reverse // DeleteCases[#, 0]&, {n, 0, 9}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

{B(n,v='x)=sum(i=0,n,binomial(n,i)*bernfrac(i)*v^(n-i))} E(n,v='x)=2/(n+1)*(B(n+1,v)-2^(n+1)*B(n+1,v/2)) \\ Ralf Stephan, Nov 05 2004

E(n, x) = 2/(n+1) * [B(n+1, x) - 2^(n+1)*B(n+1, x/2) ], with B(n, x) the Bernoulli polynomials.

Edited by Ralf Stephan, Nov 05 2004

A141684 Triangle read by rows formed from Euler polynomials: p(x,n) = if(n mod 2 = 1, 2^(1 + ((n - 1)/2))*EulerE(n, x), EulerE(n, x)); t(n,m) = Coefficients(p(x,n)).

Original entry on oeis.org

1, -1, 2, 0, -1, 1, 1, 0, -6, 4, 0, 1, 0, -2, 1, -4, 0, 20, 0, -20, 8, 0, -3, 0, 5, 0, -3, 1, 34, 0, -168, 0, 140, 0, -56, 16, 0, 17, 0, -28, 0, 14, 0, -4, 1, -496, 0, 2448, 0, -2016, 0, 672, 0, -144, 32, 0, -155, 0, 255, 0, -126, 0, 30, 0, -5, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 08 2008

Row sums are {1, 1, 0, -1, 0, 4, 0, -34, 0, 496, 0, ...}.

			{   1},
{  -1,    2},
{   0,   -1,    1},
{   1,    0,   -6,   4},
{   0,    1,    0,  -2,     1},
{  -4,    0,   20,   0,   -20,    8},
{   0,   -3,    0,   5,     0,   -3,   1},
{  34,    0, -168,   0,   140,    0, -56, 16},
{   0,   17,    0, -28,     0,   14,   0, -4,    1},
{-496,    0, 2448,   0, -2016,    0, 672,  0, -144, 32},
{   0, -155,    0, 255,     0, -126,   0, 30,    0, -5, 1}

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A060083, A004172, etc.

T[x_, n_] := If[Mod[n, 2] == 1, 2^(1 + ((n - 1)/2))*EulerE[n, x], EulerE[n, x]]; Table[Expand[T[x, n]], {n, 0, 10}]; Table[CoefficientList[T[x, n], x], {n, 0, 10}]; Flatten[%]

Edited by N. J. A. Sloane, Jan 06 2009

A171683 Triangle T(n,k) which contains 4n!2^floor((n+1)/2) times the coefficient [t^n x^k] exp(tx)/(3 + exp(2t)) in row n, column k.

Original entry on oeis.org

1, -1, 2, -1, -2, 2, 1, -6, -6, 4, 10, 4, -12, -8, 4, 26, 100, 20, -40, -20, 8, -154, 156, 300, 40, -60, -24, 8, -1646, -2156, 1092, 1400, 140, -168, -56, 16, 1000, -13168, -8624, 2912, 2800, 224, -224, -64, 16, 92744, 18000, -118512, -51744, 13104, 10080, 672, -576, -144, 32

Offset: 0

Roger L. Bagula, Dec 15 2009

The bivariate Taylor expansion of exp(t*x)/(3+exp(2*t)) is 1/4 + (x/4-1/8)*t +(-1/16+x^2/8-x/8)*t^2+...
Row n contains the coefficients of [x^k] of the polynomial in front of t^n, multiplied by 4*floor((n+1)/2)*n!.
Row sums are: 1, 1, -1, -7, -2, 94, 266, -1378, -15128, -36344, 839144,...

			The triangle starts in row n=0, columns 0<=k <=n as
      1;
     -1,      2;
     -1,     -2,       2;
      1,     -6,      -6,      4;
     10,      4,     -12,     -8,     4;
     26,    100,      20,    -40,   -20,     8;
   -154,    156,     300,     40,   -60,   -24,    8;
  -1646,  -2156,    1092,   1400,   140,  -168,  -56,   16;
   1000, -13168,   -8624,   2912,  2800,   224, -224,  -64,   16;
  92744,  18000, -118512, -51744, 13104, 10080,  672, -576, -144, 32;
  ...

Cf. A000364, A060083, A171684.

Clear[p, g, m, a];
m = 1;
p[t_] = 2^(m + 1)*Exp[t*x]/(-1 + 2^(m + 1) + Exp[2^m*t]) Table[ FullSimplify[ExpandAll[2^ Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], {n, 0, 10}]
a = Table[CoefficientList[FullSimplify[ExpandAll[2^Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}]
Flatten[a]

Number of variables in use reduced from 4 to 2, keyword:tabl added - The Assoc. Eds. of the OEIS, Oct 20 2010

cp's OEIS Frontend

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

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A102054 Triangular matrix, read by rows, where T(n,k) = T(n-1,k) - [T^-1](n-1,k-1); also equals 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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A004172 Triangle of coefficients of Euler polynomials E_2n(x) (exponents in increasing order).

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A060082 Coefficients of even-indexed Euler polynomials (falling powers without zeros).

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A141684 Triangle read by rows formed from Euler polynomials: p(x,n) = if(n mod 2 = 1, 2^(1 + ((n - 1)/2))*EulerE(n, x), EulerE(n, x)); t(n,m) = Coefficients(p(x,n)).

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A171683 Triangle T(n,k) which contains 4*n!*2^floor((n+1)/2) times the coefficient [t^n x^k] exp(t*x)/(3 + exp(2*t)) in row n, column k.

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A171683 Triangle T(n,k) which contains 4n!2^floor((n+1)/2) times the coefficient [t^n x^k] exp(tx)/(3 + exp(2t)) in row n, column k.