A110501 -id:A110501 - cp's OEIS frontend

A229023 Numerators of the main diagonal of A225825 difference table, a sequence linked to Bernoulli, Genocchi and Clausen numbers.

1, -2, 16, -424, 2944, -70240, 70873856, -212648576, 98650550272, -90228445612544, 19078660567134208, -2034677178643867648, 123160010212358914048, -19182197131374977024, 228111332170536254898176, -51166426240975948419354886144

Offset: 0

Jean-François Alcover and Paul Curtz, Sep 11 2013

a(n) is divisible by 2^n and congruent to 1, 2, 4, 5, 7 or 8 mod 9.

			1, -2/3, 16/15, -424/105, 2944/105, -70240/231, 70873856/15015, ...

Cf. A181131 (denominators), A225825, A110501 (Genocchi numbers), A141056 (Clausen numbers), A212196 (Bernoulli medians), A005439 (Genocchi medians).

nmax = 30; Clausen[n_] := Times @@ Select[Divisors[n] + 1, PrimeQ]; t = Join[{1}, Table[Numerator[BernoulliB[n, 1/2] - (n + 1)*EulerE[n, 0]]/Clausen[n], {n, 1, nmax}]]; dt = Table[Differences[t, n], {n, 0, nmax}]; Diagonal[dt] // Numerator

A296941 Expansion of e.g.f. arcsin(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, 18, 227, 12125, 542448, 55071205, 5492843269, 905996551626, 159770279801855, 39299019878991521, 10721872262093222016, 3707660329253983397113, 1438816154956071399594457, 668949924061617421125859650, 348908555505788456739965412203

Offset: 0

Ilya Gutkovskiy, Dec 22 2017

nonn

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

Cf. A001469, A001818, A012780, A110501, A296839, A296841, A296842, A296853, A296854, A296856, A296939, A296940, A296942.

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

A296942 Expansion of e.g.f. arcsinh(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, -12, -193, 5195, 397248, -9589391, -3147743231, -10931156748, 65632780196255, 4713930109297211, -2846093176389647904, -606335605925899344287, 213167747093485780707937, 109460864600185764327567060, -21782399212761670190907400897

Offset: 0

Ilya Gutkovskiy, Dec 22 2017

sign

			arcsinh(x*tan(x/2)) = x^2/2! + x^4/4! - 12*x^6/6! - 193*x^8/8! + ...

Cf. A000364, A001469, A001818, A110501, A296839, A296841, A296842, A296853, A296854, A296856, A296939, A296940, A296941.

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

A343795 Number of Dumont permutations of the fourth kind of length 2n avoiding the pattern 312.

Original entry on oeis.org

1, 1, 3, 10, 39, 174, 872, 4805, 28474, 178099, 1160173, 7803860, 53924841, 381640934, 2761331130, 20400560942, 153738854242, 1180631743440, 9229687049249, 73372263658451, 592476077260123, 4854377724124700, 40315729803287046, 339065862485375334, 2885324166565733641

Offset: 0

Alexander Burstein and Opel Jones, Apr 29 2021

Dumont permutations of the fourth kind are permutations of even length where all deficiencies (drops) are even values at even positions.

			For n=2, a(2)=3 counts the permutations 1234, 1342, 1432.

O. Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, Ph.D. thesis, Howard University, 2019.

A. Burstein and O. Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, arXiv:2002.12189 [math.CO], 2020.
A. Burstein, M. Josuat-Vergès, and W. Stromquist, New Dumont permutations, Pure Math. Appl. (Pu.M.A.) 21 (2010), no. 2, 177-206.

Cf. A000108 (permutations avoiding 312), A024492, A048990, A110501 (length 2n Dumont permutations of 4th kind).

seq(n)={my(h=sqrt(1-16*x + O(x*x^n)), F=sqrt((1-h)/(8*x)), G=(1-sqrt((1+h)/2))/(2*x), A=O(1)); forstep(k=n\3, 0, -1, my(f=Pol(F + O(x*x^k))); A = f/((1 - x*Pol(G + O(x^k)))^2 - x*f/(1 - x*Pol(G + O(x*x^k)) - x*f^2/(1 - x*A))) ); Vec(A + O(x*x^n))} \\ Andrew Howroyd, Apr 29 2021

Let F_k(x) be the truncation of the g.f. of A048990 to a polynomial of degree k. Let G_k(x) be the truncation of the g.f. of A024492 to a polynomial of degree k. Let G_{-1}(x) = 0. For k>=0, define A_k(x) recursively as follows: A_k(x) = F_k(x)/((1-x*G_{k-1}(x))^2-x*F_k(x)/(1-x*G_k(x)-x*F_k(x)^2/(1-x*A_{k+1}(x)))). Then A_0(x) is the g.f. of this sequence.

Terms a(12) and beyond from Andrew Howroyd, Apr 29 2021

A014780 Triangle of numbers associated with Genocchi numbers.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 5, 3, 3, 17, 17, 31, 25, 31, 17, 17, 155, 155, 293, 259, 349, 259, 293, 155, 155, 2073, 2073, 3991, 3681, 5151, 4289, 5151, 3681, 3991, 2073, 2073, 38227, 38227, 74381, 70235, 100325, 88507, 109765, 88507, 100325, 70235, 74381, 38227

Offset: 1

N. J. A. Sloane

When written in the triangle form as in the example below, the k-th entry (k>=1) in row n (n>=1) is the number of Dumont permutations of the first kind of length 2*n that start with k+1. - Alexander Burstein, May 14 2022

			Triangle begins:
   1;
   1,  1,  1;
   3,  3,  5,  3,  3;
  17, 17, 31, 25, 31, 17, 17;
  ...

G. Kreweras, Sur les permutations comptées par les nombres de Genocchi de 1-ière et 2-ième espèce, Europ. J. Comb., vol. 18, pp. 49-58, 1997.

Cf. A110501, A014782, A014783, A014784.

Row sums give A110501.

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 22 2003

A130653 Odd terms in A002430 = numerators in Taylor series for tan(x).

Original entry on oeis.org

1, 1, 17, 929569, 129848163681107301953, 7724760729208487305545342963324697288405380586579904269441, 357302767470032900576643605538835088084055212588960920085261795996340330997333306469144562500392344758421560010463942134842407723273904635849262137252097

Offset: 1

Alexander Adamchuk, Jun 20 2007

nonn

Odd terms in A002430(n) correspond to the indices that are the powers of 2.

			tan(x) = x + 2 x^3/3! + 16 x^5/5! + 272 x^7/7! + ... = 1*x + 1/3*x^3 + 2/15*x^5 + 17/315*x^7 + 62/2835*x^9 + O(x^10).
A002430(n) begins {1, 1, 2, 17, 62, 1382, 21844, 929569, 6404582, 443861162, 18888466084, 113927491862, 58870668456604, 8374643517010684, 689005380505609448, 129848163681107301953, ...}.
Thus a(1) = 1, a(2) = 1, a(3) = 17, a(4) = 929569, a(5) = 129848163681107301953.

Eric Weisstein's World of Mathematics, Tangent.

Cf. A002430 = Numerators in Taylor series for tan(x). Also from Taylor series for tanh(x). Cf. A001469, A002425, A046990, A089171, A110501, A036968.

Table[ Numerator[ Abs[ 2^(2^n)(2^(2^n)-1)/(2^n)! * BernoulliB[ 2^n ] ] ], {n,1,8} ]

a(n) = Numerator[ Abs[ 2^(2^n)(2^(2^n)-1)/(2^n)! * BernoulliB[ 2^n ] ] ]. a(n) = A002430(2^(n-1)).

A221094 O.g.f.: Sum_{n>=0} n!^2 * x^n / Product_{k=1..n} (1 + k(n-k+1)x).

Original entry on oeis.org

1, 1, 3, 21, 263, 5165, 146335, 5649397, 285069735, 18214525629, 1437313035887, 137272113393413, 15605422414146487, 2082375903282194893, 322303158660868063359, 57271523430269553109269, 11579903781095519639058119, 2643368434346324374530280157, 676521525314300179793917303951

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

Compare to: Sum_{n>=0} n!^2 * x^n / Product_{k=1..n} (1 + k^2*x) = Sum_{n>=0} A110501(n)*x^n, where A110501 is unsigned Genocchi numbers of even index.

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 263*x^4 + 5165*x^5 + 146335*x^6 +...
where
A(x) = 1 + x/(1+x) + 2!^2*x^2/((1+1*2*x)*(1+2*1*x)) + 3!^2*x^3/((1+1*3*x)*(1+2*2*x)*(1+3*1*x)) + 4!^2*x^4/((1+1*4*x)*(1+2*3*x)*(1+3*2*x)*(1+4*1*x)) + 5!^2*x^5/((1+1*5*x)*(1+2*4*x)*(1+3*3*x)*(1+4*2*x)*(1+5*1*x)) +...

CoefficientList[Series[Sum[(n!)^2 x^n/Product[1+k(n-k+1)x,{k,n}],{n,0,20}],{x,0,20}],x] (* Harvey P. Dale, Aug 03 2020 *)

{a(n)=polcoeff(sum(m=0, n, m!^2*x^m/prod(k=1, m, 1+(m-k+1)*k*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

A224873 Triangle of coefficients T(n, k) (n > 0, k < n) in expansion of sin(ux) sin(vx) / (cos(ux) + cos(v*x)), read by rows.

Original entry on oeis.org

1, 1, 1, 3, 25, 3, 17, 329, 329, 17, 155, 5325, 14301, 5325, 155, 2073, 110605, 563013, 563013, 110605, 2073, 38227, 2918825, 23904881, 45956625, 23904881, 2918825, 38227, 929569, 96075665, 1150348017, 3600524785, 3600524785, 1150348017, 96075665, 929569

Offset: 1

Michael Somos, Jul 23 2013

			1; 1, 1; 3, 25, 3; 17, 329, 329, 17; ...

Michael Hardy, A "known" tangent half-angle formula?.

Cf. A024235, A110501.

{T(n, k) = local(u = 'u, v = 'v, A); if( n<0 || k>=n, 0, n = 2*n; k = 2*k + 1; A = x * O(x^n); n! * polcoeff( polcoeff( polcoeff( sin( u*x + A) * sin( v*x + A) / (cos( u*x + A) + cos( v*x + A)), n, x), k, u), n - k, v))}

A024235(n) = Sum_{k = 0..n-1} T(n, k).
A110501(n) = T(n, 0).
E.g.f.: sin(u*x) * sin(v*x) / (cos(u*x) + cos(v*x)) = Sum_{n>0, k

A308750 Number of Dumont permutations of the first kind of length 2n avoiding pattern 2143 (or pattern 3421).

Original entry on oeis.org

1, 1, 2, 7, 36, 239, 1892, 17015, 168503, 1799272, 20409644

Offset: 0

Alexander Burstein and Opel Jones, Jun 21 2019

Conjecture: The number of Dumont permutations of the first kind avoiding pattern 2143 equals the number of Dumont permutations of the first kind avoiding pattern 3421 for all n >= 0.

Data for n=7,8,9,10 is due to Michael Albert.

			For n=3, the 7 Dumont permutations of the first kind avoiding pattern 2143 are 356421, 364215, 435621, 563421, 564213, 634215, 642135, and the 7 Dumont permutations of the first kind avoiding pattern 3421 are 214365, 216435, 421365, 421563, 421635, 621435, 642135.

O. Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, Ph.D. thesis, Howard University, 2019.

D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.

Cf. A001469, A110501.

A336674 Number of positive terms of the Okounkov-Olshanski formula for the number of standard tableaux of skew shape (n+3,n+2,...,1)/(n-1,n-2,...,1).

Original entry on oeis.org

1, 1, 5, 65, 1757, 87129, 7286709, 965911665, 193387756045, 56251615627273, 23021497112124901, 12903943243053179681, 9680994096074346690365, 9530338509606467082850745, 12099590059386455266220499477

Offset: 0

Alejandro H. Morales, Jul 29 2020

nonn

a(n) is also the number of semistandard Young tableaux of skew shape (n+3,n+2,...,1)/(n-1,n-2,...,1) such that the entries in row i are at most i for i=1,...,n+3.

a(n) is also the number of semistandard Young tableaux T of shape (n-1,n-2,...,1) such that j-i < T(i,j) <= n+3 for all cells (i,j).

			For n=2 the a(2)=5 semistandard Young tableaux of skew shape (5,4,3,2,1)/(1) are determined by their first column which are [1,2,3,4], [1,2,3,5], [1,2,4,5], [1,3,4,5], and [2,3,4,5]. Also, the a(2)=5 semistandard Young tableaux of shape (1) with entries between 0 and 5 are [1], [2], [3], [4], and [5]. Also, the a(3)=70-5=65 are the semistandard Young tableaux of shape (2,1) with entries at most 6 excluding the five tableaux whose entry in the first row and first column is 1: [[1,1],[2]], [[1,1],[3]], [[1,1],[4]], and [[1,1],[5]].

A. H. Morales and D. G. Zhu, On the Okounkov-Olshanski formula for standard tableaux of skew shape, arXiv:2007.05006 [math.CO], 2020.

A110501, A005700 gives the number of terms of the Naruse hook length formula for the same skew shape.

b := proc(n)
    return 2*(-1)^n*(1-4^n)*bernoulli(2*n)/factorial(2*n);
end proc:
a := proc(n)
    return factorial(2*n+4)*factorial(2*n+6)*(b(n+1)*b(n+3)-b(n+2)^2)/6;
end proc:
seq(a(n),n=0..10);

def b(n):
    return 2*(-1)^n*(1-4^n)*bernoulli(2*n)/factorial(2*n) ;
def a(n):
    return factorial(2*n+4)*factorial(2*n+6)*(b(n+1)*b(n+3)-b(n+2)^2)/6;
[a(i) for i in range(10)]

a(n) = ((2*n+4)!*(2*n+6)!/3!)*(b(n+1)*b(n+3)-b(n+2)^2) where b(n)=A110501(n)/(2*n)!.

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cp's OEIS Frontend

A229023 Numerators of the main diagonal of A225825 difference table, a sequence linked to Bernoulli, Genocchi and Clausen numbers.

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A296941 Expansion of e.g.f. arcsin(x*tan(x/2)) (even powers only).

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A296942 Expansion of e.g.f. arcsinh(x*tan(x/2)) (even powers only).

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A343795 Number of Dumont permutations of the fourth kind of length 2n avoiding the pattern 312.

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A014780 Triangle of numbers associated with Genocchi numbers.

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A130653 Odd terms in A002430 = numerators in Taylor series for tan(x).

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A221094 O.g.f.: Sum_{n>=0} n!^2 * x^n / Product_{k=1..n} (1 + k*(n-k+1)*x).

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A224873 Triangle of coefficients T(n, k) (n > 0, k < n) in expansion of sin(u*x) * sin(v*x) / (cos(u*x) + cos(v*x)), read by rows.

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A221094 O.g.f.: Sum_{n>=0} n!^2 * x^n / Product_{k=1..n} (1 + k(n-k+1)x).

A224873 Triangle of coefficients T(n, k) (n > 0, k < n) in expansion of sin(ux) sin(vx) / (cos(ux) + cos(v*x)), read by rows.