A110501 -id:A110501 - cp's OEIS frontend

A329718 The number of open tours by a biased rook on a specific f(n) X 1 board, where f(n) = A070941(n) and cells are colored white or black according to the binary representation of 2n.

1, 2, 4, 4, 8, 6, 14, 8, 16, 10, 24, 10, 46, 24, 46, 16, 32, 18, 44, 14, 84, 34, 68, 18, 146, 68, 138, 44, 230, 84, 146, 32, 64, 34, 84, 22, 160, 54, 112, 22, 276, 106, 224, 54, 376, 106, 192, 34, 454, 192, 406, 112, 690, 224, 406, 84, 1066, 376, 690, 160

Offset: 0

Mikhail Kurkov, Nov 19 2019 [verification needed]

nonn

A cell is colored white if the binary digit is 0 and a cell is colored black if the binary digit is 1. A biased rook on a white cell moves only to the left and otherwise moves only to the right.

			a(1) = 2 because the binary expansion of 2 is 10 and there are 2 open biased rook's tours, namely 12 and 21.
a(2) = 4 because the binary expansion of 4 is 100 and there are 4 open biased rook's tours, namely 132, 213, 231 and 321.
a(3) = 4 because the binary expansion of 6 is 110 and there are 4 open biased rook's tours, namely 123, 132, 231 and 312.

Mikhail Kurkov, Comments on A329718 [verification needed]

Cf. A027649, A027650, A027651, A059894, A110501, A283811, A284005, A329369, A344902, A344947.

a(n) = f(n) + f(A059894(n)) = f(n) + f(2*A053645(n)) for n > 0 with a(0) = 1 where f(n) = A329369(n).
Sum_{k=0..2^n-1} a(k) = 2*(n+1)! - 1 for n >= 0.
a((4^n-1)/3) = 2*A110501(n+1) for n > 0.
a(2^1*(2^n-1)) = A027649(n),
a(2^2*(2^n-1)) = A027650(n),
a(2^3*(2^n-1)) = A027651(n),
a(2^4*(2^n-1)) = A283811(n),
and more generally, a(2^m*(2^n-1)) = T(n,m+1) for n >= 0, m >= 0 where T(n,m) = Sum_{k=0..n} k!*(k+1)^m*Stirling2(n,k)*(-1)^(n-k).

A221371 O.g.f.: Sum_{n>=0} n!^2 * x^n * Product_{k=1..n} (1 + x) / (1 + k^2x + k^2x^2).

Original entry on oeis.org

1, 1, 4, 23, 209, 2744, 49539, 1180281, 35921892, 1360513711, 62770245601, 3466178083312, 225719029475675, 17117740162448105, 1495526385479298140, 149120758170390404103, 16831018302445533666705, 2134813624482300873515304, 302332062412598445891728563

Offset: 0

Paul D. Hanna, Jan 13 2013

nonn

Compare to the o.g.f. of A110501, the unsigned Genocchi numbers (of first kind):

Sum_{n>=0} n!^2 * x^(n+1) / Product_{k=1..n} (1 + k^2*x).

			O.g.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 209*x^4 + 2744*x^5 + 49539*x^6 + ...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + 2!^2*x^2*(1+x)*(1+x)/((1+x+x^2)*(1+4*x+4*x^2)) + 3!^2*x^3*(1+x)*(1+x)*(1+x)/((1+x+x^2)*(1+4*x+4*x^2)*(1+9*x+9*x^2)) + 4!^2*x^4*(1+x)*(1+x)*(1+x)*(1+x)/((1+x+x^2)*(1+4*x+4*x^2)*(1+9*x+9*x^2)*(1+16*x+16*x^2)) + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..240

Cf. A221370, A110501.

a[n1_Integer?NonNegative, n2_Integer?NonNegative] := CoefficientList[Sum[(m!)^2*x^m*Product[(1 + x)/(1 + k^2*x + k^2*x^2), {k, 1, m}], {m, 0, n2 + 1}] + O[x]^(n2 + 2), x][[n1 + 1 ;; n2 + 1]]; a[0, 18] (* Robert P. P. McKone, Sep 16 2023 *)

{a(n)=polcoeff( sum(m=0, n, m!^2*x^m*prod(k=1, m, (1+x)/(1+k^2*x+k^2*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=sum(k=0,n\2,binomial(n-k,k)*2*(-1)^(n-k+1)*(1-4^(n-k+1))*bernfrac(2*(n-k+1)))}
for(n=0, 30, print1(a(n), ", "))

O.g.f.: A(x) = 1/(1-x*(1+x)/(1-2*x/(1-4*x*(1+x)/(1-6*x*(1+x)/(1-9*x*(1+x)/(1-12*x*(1+x)/(... -[(n+1)/2]*[(n+2)/2]*x*(1+x)/(1- ...)))))))) (continued fraction).
a(n) = Sum_{k=0..[n/2]} binomial(n-k,k) * A110501(n+1), where A110501(n) = 2*(-1)^n*(1-4^n)*B_{2*n} (B = Bernoulli numbers).
a(n) ~ 2^(2*n+5) * n^(2*n+5/2) / (exp(2*n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, Nov 02 2014

A221972 G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (2k-1) / (1 + k(2k-1)x).

Original entry on oeis.org

1, 1, 5, 49, 797, 19417, 661829, 30067105, 1755847661, 128153307433, 11430887275733, 1223433282301681, 154741998546660605, 22833118232808363769, 3887374029443206242917, 756359660427618330221377, 166781979021653656537782029, 41372815623877107580771950025

Offset: 0

Paul D. Hanna, Feb 01 2013

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 49*x^3 + 797*x^4 + 19417*x^5 + 661829*x^6 +...
where
A(x) = 1 + x/(1+x) + 2!*1*3*x^2/((1+x)*(1+2*3*x)) + 3!*1*3*5*x^3/((1+x)*(1+2*3*x)*(1+3*5*x)) + 4!*1*3*5*7*x^4/((1+x)*(1+2*3*x)*(1+3*5*x)*(1+4*7*x)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..160

Cf. A110501, A024283.

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

a(n) = Sum_{k, 0<=k<=n} A211183(n,k)*4^(n-k). - Philippe Deléham, Feb 03 2013
G.f.: G(0) where G(k) =  1 + x*(2*k+1)*(4*k+1)/( 1 + x + 6*x*k + 8*x*k^2 - 2*x*(k+1)*(4*k+3)*(1 + x + 6*x*k + 8*x*k^2)/(2*x*(k+1)*(4*k+3) + (1 + 6*x + 14*x*k + 8*x*k^2)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 11 2013
a(n) ~ 2^(3*n+9/2) * n^(2*n+2) / (exp(2*n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, Nov 02 2014

A230682 O.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k^2 + x)/(1 + k^2*x).

Original entry on oeis.org

1, 1, 4, 21, 181, 2320, 41581, 991821, 30339364, 1156828681, 53761779721, 2990342767680, 196097039232121, 14969727522159481, 1315952342285654884, 131970189920614495581, 14974773731779775857021, 1908770813250950767227280, 271560466483540753565395621

Offset: 0

Paul D. Hanna, Oct 27 2013

nonn

Compare to an o.g.f. of Genocchi numbers of the first kind (A110501):
Sum_{n>=0} x^n * Product_{k=1..n} k^2/(1 + k^2*x).
Also, compare to a g.f. of Fibonacci numbers (A000045):
Sum_{n>=0} x^n * Product_{k=1..n} (k + x)/(1 + k*x).

			G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 181*x^4 + 2320*x^5 + 41581*x^6 +...
where
A(x) = 1 + x*(1+x)/(1+x) + x^2*(1+x)*(4+x)/((1+x)*(1+4*x)) + x^3*(1+x)*(4+x)*(9+x)/((1+x)*(1+4*x)*(1+9*x)) + x^4*(1+x)*(4+x)*(9+x)*(16+x)/((1+x)*(1+4*x)*(1+9*x)*(1+16*x)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..230

Cf. A230740.

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

a(n) ~ 2^(2*n+5) * n^(2*n+5/2) / (Pi^(2*n+3/2) * exp(2*n)). - Vaclav Kotesovec, Oct 28 2014

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

Original entry on oeis.org

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

cp's OEIS Frontend

A329718 The number of open tours by a biased rook on a specific f(n) X 1 board, where f(n) = A070941(n) and cells are colored white or black according to the binary representation of 2n.

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A221371 O.g.f.: Sum_{n>=0} n!^2 * x^n * Product_{k=1..n} (1 + x) / (1 + k^2*x + k^2*x^2).

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A221972 G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (2*k-1) / (1 + k*(2*k-1)*x).

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A230682 O.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k^2 + x)/(1 + k^2*x).

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

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Mathematica

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

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Mathematica

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

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Mathematica

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

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A221371 O.g.f.: Sum_{n>=0} n!^2 * x^n * Product_{k=1..n} (1 + x) / (1 + k^2x + k^2x^2).

A221972 G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (2k-1) / (1 + k(2k-1)x).