A001469 -id:A001469 - cp's OEIS frontend

A083014 a(n) = Sum_{k=0..n-1} 10^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.

0, 1, -9, 36, 81, -1524, -4779, 155316, 643761, -28041444, -145069299, 7794224196, 48371836041, -3078058903764, -22284938832219, 1637087002046676, 13545357290061921, -1127884947406124484, -10498665795419017539, 977073296798704710756

Offset: 0

Benoit Cloitre, May 31 2003

sign

Seiichi Manyama, Table of n, a(n) for n = 0..417
Ira M. Gessel, On the Almkvist-Meurman Theorem for Bernoulli Polynomials, Integers (2023) Vol. 23, #A14.

Cf. A001469.

Cf. A036968, A083007, A083008, A083009, A083010, A083011, A083012, A083013.

Range[0, 15]! CoefficientList[ Series[ 10x/(1 + Exp[x] + Exp[ 2x] + Exp[ 3x] + Exp[ 4x] + Exp[ 5x] + Exp[ 6x] + Exp[ 7x] + Exp[ 8x] + Exp[ 9x]), {x, 0, 15}], x] (* Robert G. Wilson v, Oct 26 2012 *)
Array[Sum[10^k*BernoulliB[k]*Binomial[#, k], {k, 0, # - 1}] &, 20, 0] (* Michael De Vlieger, Feb 14 2023 *)

a(n)=sum(k=0,n-1,10^k*binomial(n,k)*bernfrac(k))

E.g.f.: 10*x/(Sum_{i=0..9} exp(i*x)). - Alois P. Heinz, Sep 28 2016

Offset changed to 0 by Seiichi Manyama, Sep 28 2016

A296841 Expansion of e.g.f. sin(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, -12, -193, -2365, -18552, 500689, 48649969, 2981261772, 169237306055, 9187565146331, 427287357700176, 6011297159973313, -2887128048794477663, -711942625068679870620, -132369975517302093882097, -22968753773651295426439021

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			sin(x*tan(x/2)) = x^2/2! + x^4/4! - 12*x^6/6! - 193*x^8/8! - 2365*x^10/10! - 18552*x^12/12! + ...

Cf. A001469, A003706, A009515, A110501, A296839, A296842.

nmax = 17; Table[(CoefficientList[Series[Sin[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] sin(x*tan(x/2)).

A296842 Expansion of e.g.f. cos(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 0, -3, -15, -14, 1755, 60357, 1740284, 45816165, 776485557, -37342503290, -7203185712261, -822818831400759, -85463040449605000, -8640073895507612019, -843669753827174738535, -73050419139737972150438, -3478007209663880122501701

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			cos(x*tan(x/2)) = 1 - 3*x^4/4! - 15*x^6/6! - 14*x^8/8! + 1755*x^10/10! + 60357*x^12/12! + ...

Cf. A001469, A003710, A009080, A110501, A296839, A296841.

nmax = 17; Table[(CoefficientList[Series[Cos[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] cos(x*tan(x/2)).

A002451 Expansion of 1/((1-x)(1-4x)(1-9x)).

Original entry on oeis.org

1, 14, 147, 1408, 13013, 118482, 1071799, 9668036, 87099705, 784246870, 7059619931, 63542171784, 571901915677, 5147206719578, 46325218390143, 416928397167052, 3752361301126529, 33771274616631006, 303941563175648035, 2735474435084708240, 24619271381777877861

Offset: 0

N. J. A. Sloane

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 35.

G. C. Greubel, Table of n, a(n) for n = 0..1000
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (14,-49,36).

A diagonal of A036969.

List([0..30],n->1/24-4^(n+2)/15+9^(n+2)/40); # Muniru A Asiru, Dec 18 2018

[(10 - 4^(n+4) +6*9^(n+2))/240: n in [0..30]]; // G. C. Greubel, Jul 04 2019

a:=n->sum((9^(n-j)-4^(n-j))/5,j=0..n): seq(a(n), n=1..30); # Zerinvary Lajos, Jan 15 2007
A002451:=-1/(z-1)/(4*z-1)/(9*z-1); # Simon Plouffe in his 1992 dissertation

CoefficientList[Series[1/((1-x)(1-4x)(1-9x)), {x, 0, 30}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)

Vec(1/((1-x)*(1-4*x)*(1-9*x))+O(x^30)) \\ Charles R Greathouse IV, Sep 23 2012

[(10 - 4^(n+4) +6*9^(n+2))/240 for n in (0..30)] # G. C. Greubel, Jul 04 2019

a(n) = 1/24 - 4^(n+2)/15 + 9^(n+2)/40. - Antonio Alberto Olivares, Feb 03 2010

a(n) = 13*a(n-1) - 36*a(n-2) + 1, n >= 2. - Vincenzo Librandi, Mar 23 2011

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

A296853 Expansion of e.g.f. tanh(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, -27, -403, 8345, 688473, -208019, -3189211931, -162605047455, 28806493001105, 5257860587364341, -288068264497990179, -230932276247139756887, -14420179324444754436023, 13944106915630111553887485, 3643613240568912544562868053

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			tanh(x*tan(x/2)) = x^2/2! + x^4/4! - 27*x^6/6! - 403*x^8/8! + 8345*x^10/10! + ...

Cf. A000182, A001469, A003721, A009817, A110501, A296839, A296841, A296842, A296854, A296856.

nmax = 16; Table[(CoefficientList[Series[Tanh[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] tanh(x*tan(x/2)).

A296854 Expansion of e.g.f. sinh(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, 18, 227, 4565, 126648, 4620805, 213569269, 12165013026, 835868220455, 68093897815361, 6483538063860336, 712877916658802713, 89586864207214060057, 12753583150716684461970, 2040805972702652020364603, 364567588100855831300341565

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

			sinh(x*tan(x/2)) = x^2/2! + x^4/4! + 18*x^6/6! + 227*x^8/8! + 4565*x^10/10! + ...

Cf. A001469, A003717, A009611, A110501, A296839, A296841, A296842, A296853, A296856.

nmax = 17; Table[(CoefficientList[Series[Sinh[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] sinh(x*tan(x/2)).

A296856 Expansion of e.g.f. cosh(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 0, 3, 15, 224, 4545, 126753, 4626076, 213703095, 12167727543, 835893746300, 68091766034061, 6483302813035857, 712860388963255000, 89585739948801890619, 12753524767335858733935, 2040804997678590563632568, 364567987004433619078313961

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

			cosh(x*tan(x/2)) = 1 + 3*x^4/4! + 15*x^6/6! + 224*x^8/8! + 4545*x^10/10! + 126753*x^12/12! + ...

Cf. A001469, A003711, A009166, A110501, A296839, A296841, A296842, A296853, A296854.

nmax = 17; Table[(CoefficientList[Series[Cosh[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] cosh(x*tan(x/2)).

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A083014 a(n) = Sum_{k=0..n-1} 10^k*B(k)*binomial(n,k) where B(k) is the k-th Bernoulli number.

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A296841 Expansion of e.g.f. sin(x*tan(x/2)) (even powers only).

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A296842 Expansion of e.g.f. cos(x*tan(x/2)) (even powers only).

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A002451 Expansion of 1/((1-x)*(1-4*x)*(1-9*x)).

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

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A296853 Expansion of e.g.f. tanh(x*tan(x/2)) (even powers only).

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A296854 Expansion of e.g.f. sinh(x*tan(x/2)) (even powers only).

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A296856 Expansion of e.g.f. cosh(x*tan(x/2)) (even powers only).

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A083014 a(n) = Sum_{k=0..n-1} 10^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.

A002451 Expansion of 1/((1-x)(1-4x)(1-9x)).