A001469 -id:A001469 - cp's OEIS frontend

A296835 Expansion of e.g.f. exp(x*tan(x/2)) (even powers only).

1, 1, 4, 33, 451, 9110, 253401, 9246881, 427272364, 24332740569, 1671761966755, 136185663849422, 12966840876896193, 1425738305622057713, 179172604156015950676, 25507107918052543195905, 4081610970381242583997171, 729135575105289450378655526

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

			exp(x*tan(x/2)) = 1 + x^2/2! + 4*x^4/4! + 33*x^6/6! + 451*x^8/8! + ...

Cf. A001469, A003723, A006229, A009252, A009273, A110501, A296836.

nmax = 17; Table[(CoefficientList[Series[Exp[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)] exp(x*tan(x/2)).

A296837 Expansion of e.g.f. log(1 + x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, -2, 18, -312, 9470, -436860, 28616322, -2522596496, 288046961190, -41355026494020, 7291524732108650, -1548849359704927896, 390122366308850972238, -114968364853645904762252, 39189956630839558368115410, -15300235972710835734174638880

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			log(1 + x*tan(x/2)) = x^2/2! - 2*x^4/4! + 18*x^6/6! - 312*x^8/8! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A001469, A003707, A009379, A009399, A110501, A296838.

nmax = 16; Table[(CoefficientList[Series[Log[1 + 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)] log(1 + x*tan(x/2)).

a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = 1.54340463841820844795870974005331555369788376471926269... is the root of the equation r*tanh(r/2) = 1. - Vaclav Kotesovec, Dec 21 2017

A296838 Expansion of e.g.f. log(1 + x*tanh(x/2)) (even powers only).

Original entry on oeis.org

0, 1, -4, 48, -1186, 50060, -3226206, 294835184, -36270477034, 5779302944436, -1157856177719830, 284876691727454552, -84442374415240892898, 29680054107768128647388, -12205478262363331593956686, 5805823539844285054558025280, -3163004294186696659107788567386

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			log(1 + x*tanh(x/2)) = x^2/2! - 4*x^4/4! + 48*x^6/6! - 1186*x^8/8! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A001469, A003707, A009379, A009399, A110501, A296837.

nmax = 16; Table[(CoefficientList[Series[Log[1 + x Tanh[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)] log(1 + x*tanh(x/2)).

a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = 1.306542374188806202228727831923118284841279755635... is the root of the equation r * tan(r/2) = 1. - Vaclav Kotesovec, Dec 21 2017

A296939 Expansion of e.g.f. sec(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 0, 3, 15, 644, 17145, 1124673, 74115496, 7730031915, 921044459943, 145334164141820, 26830525240048761, 6053646614467427553, 1586816790903080698000, 487642998132913180824819, 171640559783810345998524735, 69078935661419038650738789428

Offset: 0

Ilya Gutkovskiy, Dec 22 2017

nonn

			sec(x*tan(x/2)) = 1 + 3*x^4/4! + 15*x^6/6! + 644*x^8/8! + ...

Cf. A000364, A001469, A009010, A110501, A296839, A296841, A296842, A296853, A296854, A296856, A296940.

nmax = 16; Table[(CoefficientList[Series[Sec[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)] sec(x*tan(x/2)).

A296940 Expansion of e.g.f. sech(x*tan(x/2)) (even powers only).

Original entry on oeis.org

1, 0, -3, -15, 406, 14355, -189123, -42283696, -837846615, 284972761557, 28521503291230, -3070544172379761, -1054107683427761463, 1143265731049052000, 54900209444888714822181, 7959249060310612253252265, -3679623847504649619798598778, -1631286181830482909037469295781

Offset: 0

Ilya Gutkovskiy, Dec 22 2017

sign

			sech(x*tan(x/2)) = 1 - 3*x^4/4! - 15*x^6/6! + 406*x^8/8! + ...

Cf. A000364, A001469, A009011, A110501, A296839, A296841, A296842, A296853, A296854, A296856, A296939.

nmax = 17; Table[(CoefficientList[Series[Sech[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)] sech(x*tan(x/2)).

A297703 The Genocchi triangle read by rows, T(n,k) for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 8, 14, 17, 17, 56, 104, 138, 155, 155, 608, 1160, 1608, 1918, 2073, 2073, 9440, 18272, 25944, 32008, 36154, 38227, 38227, 198272, 387104, 557664, 702280, 814888, 891342, 929569, 929569, 5410688, 10623104, 15448416, 19716064, 23281432, 26031912

Offset: 0

Peter Luschny, Jan 03 2018

			The triangle starts:
0: [     1]
1: [     1,      1]
2: [     2,      3,      3]
3: [     8,     14,     17,     17]
4: [    56,    104,    138,    155,    155]
5: [   608,   1160,   1608,   1918,   2073,   2073]
6: [  9440,  18272,  25944,  32008,  36154,  38227,  38227]
7: [198272, 387104, 557664, 702280, 814888, 891342, 929569, 929569]

Row sums are A005439 with offset 0.
T(n,0) = A005439 with A005439(0) = 1.
T(n,n) = A110501 with offset 0.
Cf. A001469, A014781, A099959, A226158.

function A297703Triangle(len::Int)
    A = fill(BigInt(0), len+2); A[2] = 1
    for n in 2:len+1
        for k in n:-1:2 A[k] += A[k+1] end
        for k in 2: 1:n A[k] += A[k-1] end
        println(A[2:n])
    end
end
println(A297703Triangle(9))

from functools import cache
@cache
def T(n):  # returns row n
    if n == 0: return [1]
    row = [0] + T(n - 1) + [0]
    for k in range(n, 0, -1): row[k] += row[k + 1]
    for k in range(2, n + 2): row[k] += row[k - 1]
    return row[1:]
for n in range(9): print(T(n))  # Peter Luschny, Jun 03 2022

A065747 Triangle of Gandhi polynomial coefficients.

Original entry on oeis.org

1, 1, 3, 3, 7, 30, 51, 42, 15, 145, 753, 1656, 1995, 1410, 567, 105, 6631, 39048, 100704, 149394, 140475, 86562, 34566, 8316, 945, 566641, 3656439, 10546413, 17972598, 20133921, 15581349, 8493555, 3246642, 841239, 135135, 10395

Offset: 1

Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 16 2001

First column is A064624.

			Triangle starts
1;
1, 3, 3;
7, 30, 51, 42, 15;
145, 753, 1656, 1995, 1410, 567, 105;
6631 ...

Michael Domaratzki, Combinatorial Interpretations of a Generalization of the Genocchi Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6.
D. Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, (in French), Discrete Mathematics 1 (1972) 321-327.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)

Cf. A036970, A064624, A065748

Let B(X, n) = X^3 (B(X+1, n-1) - B(X, n-1)), B(X, 1) = X^3; then the (i, j)-th entry is the table is the coefficient of X^(2+j) in B(X, i).

A065756 Generalization of the Genocchi numbers given by the Gandhi polynomials A(n+1,r) = r^5 A(n, r + 1) - (r - 1)^5 A(n, r); A(1,r) = r^5 - (r-1)^5.

Original entry on oeis.org

1, 1, 31, 6721, 5850271, 15060446401, 94396946822431, 1258620297379341121, 32323181593821704288671, 1481630482369728860007652801, 114129022540066183425609121804831

Offset: 1

Mike Domaratzki (mdomaratzki(AT)alumni.uwaterloo.ca), Nov 17 2001

M. Domaratzki, A Generalization of the Genocchi Numbers with Applications to Enumeration of Finite Automata, Technical Report 2001-449, Department of Computing and Information Science, Queen's University at Kingston (Kingston, Canada).
Michael Domaratzki, Combinatorial Interpretations of a Generalization of the Genocchi Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6.
D. Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, (in French), Discrete Mathematics 1 (1972) 321-327.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)

Cf. A001469, A065755, A065757, A064624, A064625.

a[n_ /; n >= 0, r_ /; r >= 0] := a[n, r] = r^5*a[n-1, r+1]-(r-1)^5*a[n-1, r]; a[1, r_ /; r >= 0] := r^5-(r-1)^5; a[, ] = 1; a[n_] := a[n-1, 1]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, May 23 2013 *)

a(n) = A(n-1, 1) for the above Gandhi polynomials.

A240485 a(n) = -Zeta(1-n)n(2^(n+1) - 4) - Zeta(-n)(n+1)(2^(n+2) - 2), for n = 0 the limit is understood.

Original entry on oeis.org

1, 3, 2, -1, -2, 3, 6, -17, -34, 155, 310, -2073, -4146, 38227, 76454, -929569, -1859138, 28820619, 57641238, -1109652905, -2219305810, 51943281731, 103886563462, -2905151042481, -5810302084962, 191329672483963, 382659344967926, -14655626154768697

Offset: 0

Paul Curtz, Apr 06 2014

sign

Let G(m, n) denote the difference table of a(n):
1,    3,   2,  -1,  -2,   3,   6, -17, -34,...
2,   -1,  -3,  -1,   5,   3, -23, -17,...
-3,  -2,   2,   6,  -2, -26,   6,...
1,    4,   4,  -8, -24,  32,...
3,    0, -12, -16,  56,...
-3, -12,  -4,  72,...
-9,   8,  76,...
17,  68,...
51,...
a(n) = G(0, n).
The main diagonal G(n, n) = 1, -1, 2, -8, 56, -608,... is essentially a signed version of A005439.
The first upper diagonal is the main diagonal multiplied by 3. G(n, n+1) = 3*G(n, n).
G(m, n) = G(m, n-1) + G(m+1,n-1).
Inverse binomial transform: after 1, 2, -3, A110501(n+1) is interleaved with 3*A110501(n+1), signed two by two. I. e. b(n) = 1, 2, -3, 1, 3, -3, -9, 17, 51,... . a(n+2) + b(n+2) = -1, 0, 1, 0, -3, 0, 17,... = A226158(n+2).
This is particular to the Genocchi numbers. If the first upper diagonal is proportional to the main diagonal (1, -1, 2, -8,...), the sequence and the inverse binomial transform are simply connected to the Genocchi numbers.

Cf. A001469, A005439, A110501, A226158, A230324.

A240485 := proc(n) if n = 0 then 1 elif n = 1 then 3 else
m := 2*iquo(n-1, 2) + 2; -2^irem(n-1, 2)*m*euler(m-1, 0) fi end:
seq(A240485(n), n=0..27); # Peter Luschny, Apr 09 2014

a[n_] := Which[n == 0, 1, n == 1, 3, True, m = 2*Quotient[n-1, 2]+2; -2^Mod[n-1, 2]*m*EulerE[m-1, 0]]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Apr 09 2014, after Peter Luschny *)

def A240485(n):
    if n < 3: return [1,3,2][n]
    m = 2*((n+1)//2)
    b = 2*(1-2^m)*bernoulli(m)
    if is_even(n): b = 2*b
    return (-1)^ceil((n^2+1)/2)*b
[A240485(n) for n in (0..24)]  # Peter Luschny, Apr 08 2014

a(2*n+1) = a(2*n+2)/2 for n > 0.
-a(2*n+2)/2 = A226158(2*n+2) = A001469(n+1) = (2*n+2)*E(2*n+1, 0) where E(n, x) are the Euler polynomials.
a(n) = -2*A226158(n) - A226158(n+1).
E.g.f.: (2*exp(x)*(3*x+exp(x)*(2*x+1)+1))/(exp(x)+1)^2. - Peter Luschny, Apr 10 2014

A263445 a(n) = (2n+1)(n+1)!Bernoulli(2n).

Original entry on oeis.org

1, 1, -1, 4, -36, 600, -16584, 705600, -43751232, 3790108800, -443539877760, 68218849036800, -13478425925184000, 3355402067989171200, -1035218714714606822400, 390189256983139461120000, -177430554756972746695065600, 96269372301568677170319360000

Offset: 0

Vladimir Reshetnikov, Oct 18 2015

sign

Robert Israel, Table of n, a(n) for n = 0..210
Index entries for sequences related to Bernoulli numbers.

Bernoulli numbers are A000367/A002445. Cf. A004193, A001332, A000182, A001469.

seq((2*n+1)*(n+1)!*bernoulli(2*n), n=0..50); # Robert Israel, Oct 18 2015

Table[(2n + 1) (n + 1)! BernoulliB[2n], {n, 0, 17}]

vector(30, n, n--; (2*n+1)*(n+1)!*bernfrac(2*n)) \\ Altug Alkan, Oct 18 2015

from math import factorial
from sympy import bernoulli
def A263445(n): return (2*n+1)*factorial(n+1)*bernoulli(2*n) # Chai Wah Wu, May 18 2022

a(n) = (2n+1)*(n+1)!*Bernoulli(2n).

a(n) ~ (-1)^(n+1)*8*sqrt(2)*n^3*(n/e)^(3*n)*Pi^(1-2*n). - Vladimir Reshetnikov, Sep 05 2016

cp's OEIS Frontend

A296835 Expansion of e.g.f. exp(x*tan(x/2)) (even powers only).

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A296837 Expansion of e.g.f. log(1 + x*tan(x/2)) (even powers only).

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A296838 Expansion of e.g.f. log(1 + x*tanh(x/2)) (even powers only).

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A296939 Expansion of e.g.f. sec(x*tan(x/2)) (even powers only).

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A296940 Expansion of e.g.f. sech(x*tan(x/2)) (even powers only).

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A297703 The Genocchi triangle read by rows, T(n,k) for n>=0 and 0<=k<=n.

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A065747 Triangle of Gandhi polynomial coefficients.

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A065756 Generalization of the Genocchi numbers given by the Gandhi polynomials A(n+1,r) = r^5 A(n, r + 1) - (r - 1)^5 A(n, r); A(1,r) = r^5 - (r-1)^5.

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A240485 a(n) = -Zeta(1-n)*n*(2^(n+1) - 4) - Zeta(-n)*(n+1)*(2^(n+2) - 2), for n = 0 the limit is understood.

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A240485 a(n) = -Zeta(1-n)n(2^(n+1) - 4) - Zeta(-n)(n+1)(2^(n+2) - 2), for n = 0 the limit is understood.

A263445 a(n) = (2n+1)(n+1)!Bernoulli(2n).