A001818 -id:A001818 - cp's OEIS frontend

A290770 a(n) = Product_{k=1..n} k^(2*k).

1, 1, 16, 11664, 764411904, 7464960000000000, 16249593066946560000000000, 11020848942410302096869949440000000000, 3102093199396597590886754340698424229232640000000000, 465607547420733489126893933985879279492195953053596584509440000000000

Offset: 0

Ilya Gutkovskiy, Aug 10 2017

G. C. Greubel, Table of n, a(n) for n = 0..27
Eric Weisstein's World of Mathematics, Hyperfactorial
Index entries for sequences related to factorial numbers

Cf. A000178, A001044, A001818, A002109, A051675, A055209, A061742, A062206, A074962, A184877, A185141, A260122.

[1] cat [(&*[k^(2*k): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

Table[Product[k^(2 k), {k, 1, n}], {n, 0, 9}]
Table[Hyperfactorial[n]^2, {n, 0, 9}]
Table[n!^(2 n)/BarnesG[n + 1]^2, {n, 0, 9}]

a(n) = prod(k=1, n, k^(2*k)) \\ Felix Fröhlich, Aug 10 2017

a(n) = A002109(n)^2.
a(n) = A185141(n)/A000178(n-1)^2 for n > 0.
a(n) = (n!)^(2*n)/G(n+1)^2, where G() is the Barnes G-function.
a(n) ~ A^2*exp(-n^2/2)*n^(n*(n+1))*n^(1/6), where A is the Glaisher-Kinkelin constant (A074962).

A291483 Expansion of e.g.f. arcsinh(x)*exp(x).

Original entry on oeis.org

0, 1, 2, 2, 0, 4, 40, -64, -1344, 3984, 85408, -356896, -8462080, 45908160, 1209040768, -8080805888, -235449260032, 1871655631104, 59955521585664, -552758145525248, -19339870285225984, 202927333558572032, 7707208199780517888, -90698934927786770432, -3718489569130941169664, 48507735629457304555520

Offset: 0

Ilya Gutkovskiy, Aug 24 2017

sign

			E.g.f.: A(x) = x/1! + 2*x^2/2! + 2*x^3/3! + 4*x^5/5! + 40*x^6/6! - 64*x^7/7! - 1344*x^8/8! + ...

Cf. A001818, A009545, A012584, A131577, A291482.

a:=series(arcsinh(x)*exp(x),x=0,26): seq(n!*coeff(a,x,n),n=0..25); # Paolo P. Lava, Mar 27 2019

nmax = 25; Range[0, nmax]! CoefficientList[Series[ArcSinh[x] Exp[x], {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[Log[x + Sqrt[1 + x^2]] Exp[x], {x, 0, nmax}], x]
nmax = 25; Range[0, nmax]! CoefficientList[Series[-Sum[((-1)^k (-1 + x + Sqrt[1 + x^2])^k)/k, {k, 1, Infinity}] Exp[x], {x, 0, nmax}], x]

E.g.f.: log(x + sqrt(1 + x^2))*exp(x).

A291527 E.g.f. A(x,k) satisfies: sn(A(x,k), k) = k * sn(x,k), where sn(,) and cn(,) are Jacobi Elliptic functions.

Original entry on oeis.org

1, -1, 0, 1, 1, 4, -10, -4, 9, -1, -44, 75, 224, -299, -180, 225, 1, 408, 92, -7400, 4758, 19592, -15876, -12600, 11025, -1, -3688, -23387, 194160, 155702, -1313312, 264586, 2445840, -1289925, -1323000, 893025, 1, 33212, 804210, -3980044, -20402105, 64915224, 74573980, -279362392, -18229761, 414859500, -144802350, -196465500, 108056025, -1, -298932, -22347185, 33998224, 1349961795, -1942776004, -12484642765, 21458573952, 32679754381, -72263858940, -19224079875, 92046754800, -20560114575, -39332393100, 18261468225, 1, 2690416, 581249144, 2783246128, -71371497796, -59230867280, 1313526021896, -606679979408, -7350770598874, 7512502827344, 15289334428104, -22529210886000, -9997446759300, 25906255174800, -3292683193800, -10226422206000, 4108830350625

Offset: 1

Paul D. Hanna, Aug 25 2017

Compare to the law of sines of a spherical triangle: sin(A)/sin(a) = k.

The series reversion of e.g.f. A(x,k) wrt x equals A(k*x, 1/k) / k.

			This irregular triangle of coefficients T(n,r) in A(x,k) begins:
[1],
[-1, 0, 1],
[1, 4, -10, -4, 9],
[-1, -44, 75, 224, -299, -180, 225],
[1, 408, 92, -7400, 4758, 19592, -15876, -12600, 11025],
[-1, -3688, -23387, 194160, 155702, -1313312, 264586, 2445840, -1289925, -1323000, 893025],
[1, 33212, 804210, -3980044, -20402105, 64915224, 74573980, -279362392, -18229761, 414859500, -144802350, -196465500, 108056025],
[-1, -298932, -22347185, 33998224, 1349961795, -1942776004, -12484642765, 21458573952, 32679754381, -72263858940, -19224079875, 92046754800, -20560114575, -39332393100, 18261468225],
[1, 2690416, 581249144, 2783246128, -71371497796, -59230867280, 1313526021896, -606679979408, -7350770598874, 7512502827344, 15289334428104, -22529210886000, -9997446759300, 25906255174800, -3292683193800, -10226422206000, 4108830350625], ...
where e.g.f. A(x,k) = Sum_{n>=1, r=1..2*n-1} T(n,r) * x^(2*n-1) * k^(2*r-1) / (2*n-1)!.
E.g.f.: A(x,k) = k*x + (k^5 - k)*x^3/3! +
(9*k^9 - 4*k^7 - 10*k^5 + 4*k^3 + k)*x^5/5! +
(225*k^13 - 180*k^11 - 299*k^9 + 224*k^7 + 75*k^5 - 44*k^3 - k)*x^7/7! +
(11025*k^17 - 12600*k^15 - 15876*k^13 + 19592*k^11 + 4758*k^9 - 7400*k^7 + 92*k^5 + 408*k^3 + k)*x^9/9! +
(893025*k^21 - 1323000*k^19 - 1289925*k^17 + 2445840*k^15 + 264586*k^13 - 1313312*k^11 + 155702*k^9 + 194160*k^7 - 23387*k^5 - 3688*k^3 - k)*x^11/11! +
(108056025*k^25 - 196465500*k^23 - 144802350*k^21 + 414859500*k^19 - 18229761*k^17 - 279362392*k^15 + 74573980*k^13 + 64915224*k^11 - 20402105*k^9 - 3980044*k^7 + 804210*k^5 + 33212*k^3 + k)*x^13/13! +
(18261468225*k^29 - 39332393100*k^27 - 20560114575*k^25 + 92046754800*k^23 - 19224079875*k^21 - 72263858940*k^19 + 32679754381*k^17 + 21458573952*k^15 - 12484642765*k^13 - 1942776004*k^11 + 1349961795*k^9 + 33998224*k^7 - 22347185*k^5 - 298932*k^3 - k)*x^15/15! +...
such that
(1) sn(A(x,k), k) = k * sn(x,k),
(2) cn(A(x,k), k) = dn(x,k),
(3) dn(A(k*x,1/k)/k, k) = cn(x,k),
(4) A(k * A(x,k), 1/k) = k * x,
(5) A(A(x,1/k) / k, k) = x / k.
RELATED SERIES.
Let A^r(x,k) denote the r-th iteration of A(x,k) wrt x, then
sn( A^r(x,k), k) = k^r * sn(x,k).
For example, sn( A(A(x,k), k), k) = k^2 * sn(x,k), where
A(A(x,k), k) = k^2*x + (k^8 + k^6 - k^4 - k^2)*x^3/3! + (9*k^14 + 6*k^12 - k^10 - 20*k^8 - 9*k^6 + 14*k^4 + k^2)*x^5/5! + (225*k^20 + 135*k^18 - 180*k^16 - 300*k^14- 434*k^12 + 210*k^10 + 524*k^8 - 44*k^6 - 135*k^4 - k^2)*x^7/7! + (11025*k^26 + 6300*k^24 - 13230*k^22 - 23940*k^20 - 2961*k^18 + 6552*k^16 + 18332*k^14 + 22712*k^12 - 17825*k^10 - 12852*k^8 + 4658*k^6 + 1228*k^4 + k^2)*x^9/9! + (893025*k^32 + 496125*k^30 - 1393875*k^28 - 2433375*k^26 - 335475*k^24 + 3138345*k^22 + 866745*k^20 - 82995*k^18 + 562771*k^16 - 2154361*k^14 - 783465*k^12 + 1194707*k^10 + 201343*k^8 - 158445*k^6 - 11069*k^4 - k^2)*x^11/11! +...
Related Jacobi elliptic functions sn(,), cn(,), and dn(,) begin:
sn(x,k) = x + (-k^2 - 1)*x^3/3! + (k^4 + 14*k^2 + 1)*x^5/5! + (-k^6 - 135*k^4 - 135*k^2 - 1)*x^7/7! + (k^8 + 1228*k^6 + 5478*k^4 + 1228*k^2 + 1)*x^9/9! + (-k^10 - 11069*k^8 - 165826*k^6 - 165826*k^4 - 11069*k^2 - 1)*x^11/11! + (k^12 + 99642*k^10 + 4494351*k^8 + 13180268*k^6 + 4494351*k^4 + 99642*k^2 + 1)*x^13/13! + (-k^14 - 896803*k^12 - 116294673*k^10 - 834687179*k^8 - 834687179*k^6 - 116294673*k^4 - 896803*k^2 - 1)*x^15/15! +...
where sn(x,k) = sn(A(x,k), k)/k.
cn(x,k) = 1 - x^2/2! + (4*k^2 + 1)*x^4/4! + (-16*k^4 - 44*k^2 - 1)*x^6/6! + (64*k^6 + 912*k^4 + 408*k^2 + 1)*x^8/8! + (-256*k^8 - 15808*k^6 - 30768*k^4 - 3688*k^2 - 1)*x^10/10! + (1024*k^10 + 259328*k^8 + 1538560*k^6 + 870640*k^4 + 33212*k^2 + 1)*x^12/12! + (-4096*k^12 - 4180992*k^10 - 65008896*k^8 - 106923008*k^6 - 22945056*k^4 - 298932*k^2 - 1)*x^14/14! +...
where cn(x,k) = dn(A(k*x,1/k)/k, k),
and cn(2*A(x,k), k) = -1 + 2*dn(x,k)^2 / (1 - k^6*sn(x,k)^4).
dn(x,k) = 1 - k^2*x^2/2! + (k^4 + 4*k^2)*x^4/4! + (-k^6 - 44*k^4 - 16*k^2)*x^6/6! + (k^8 + 408*k^6 + 912*k^4 + 64*k^2)*x^8/8! + (-k^10 - 3688*k^8 -30768*k^6 - 15808*k^4 - 256*k^2)*x^10/10! + (k^12 + 33212*k^10 + 870640*k^8 + 1538560*k^6 + 259328*k^4 + 1024*k^2)*x^12/12! + (-k^14 - 298932*k^12 - 22945056*k^10 - 106923008*k^8 - 65008896*k^6 - 4180992*k^4 - 4096*k^2)*x^14/14! +...
where dn(x,k) = cn(A(x,k),k).

Paul D. Hanna, Table of n, a(n) for n = 1..900 of rows 1..30 read as a flattened table.

Cf. A060627, A002754, A060628, A001818.

Cf. A291560.

/* Find A such that sn(A,k) = k * sn(x,k) */
{T(n,r) = my(A=x,V=[k],S=x,C=1-x^2/2);
for(m=0,n, V=concat(V,[0,0]); A = x*Ser(V);
S = intformal(C*subst(C,x,A));
C = 1 - intformal(S*subst(C,x,A));
V[#V] = -polcoeff(subst(S,x,A)/S,#V-1,x););
(2*n-1)!*polcoeff(V[2*n-1],2*r-1,k)}
for(n=1,10, for(r=1,2*n-1, print1(T(n,r),", "));print(""))

{T(n, k) = my(A, m); if( n<0 || k>=(m=2*n+1), 0, A = intformal(1 / sqrt((1 - x^2) * (1 - y^2*x^2) + x*O(x^m))); A = subst(A, x, y * serreverse(A)); m! * polcoeff( polcoeff(A, m), 2*k+1))}; /* Michael Somos, Aug 27 2017 */

E.g.f. A(x,k) = Sum_{n>=1, r=1..2*n-1} T(n,r) * x^(2*n-1) * k^(2*r-1)/(2*n-1)!, satisfies:
(1) sn(A(x,k), k) = k * sn(x,k),
(2) cn(A(x,k), k) = dn(x,k),
(3) dn(A(k*x,1/k)/k, k) = cn(x,k),
(4) A(k*A(x,k), 1/k) = k*x,
(5) A(A(x,1/k)/k, k) = x/k,
(6) sn( A^r(x,k), k) = k^r * sn(x,k) where A^r(x,k) = A( A^{r-1}(x,k), k) is the r-th iteration of A(x,k) wrt x, with A^0(x,k) = x.
Row sums of n-th row equals zero for n>1.
T(n+1,1) = (-1)^n for n>=0.
T(n+1, 2*n+1) = ( (2*n)! / (n!*2^n) )^2 = A001818(n) for n>=0.

A296677 Expansion of e.g.f. arctan(arcsin(x)) (odd powers only).

Original entry on oeis.org

1, -1, 13, -173, 12409, -370137, 88556037, -2668274373, 2491377242481, 34526890553679, 202383113207336829, 25792743610973373219, 39172126704113226631401, 12501799823936578879327095, 15717805122762984314778029685, 9078237580992214462785729689355

Offset: 0

Ilya Gutkovskiy, Dec 18 2017

sign

			arctan(arcsin(x)) = x/1! - x^3/3! + 13*x^5/5! - 173*x^7/7! + 12409*x^9/9! - 370137*x^11/11! + ...

Cf. A001818, A003705, A010050, A012194, A096719, A096720, A296465, A296467.

nmax = 16; Table[(CoefficientList[Series[ArcTan[ArcSin[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 16; Table[(CoefficientList[Series[(I/2) Log[1 - Log[I x + Sqrt[1 - x^2]]] - (I/2) Log[1 + Log[I x + Sqrt[1 - x^2]]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

E.g.f.: (i/2)*log(1 - log(i*x + sqrt(1 - x^2))) - (i/2)*log(1 + log(i*x + sqrt(1 - x^2))), where i is the imaginary unit (odd powers only).

A296726 Expansion of e.g.f. arcsin(x)/(1 - x).

Original entry on oeis.org

0, 1, 2, 7, 28, 149, 894, 6483, 51864, 477801, 4778010, 53451135, 641413620, 8446433085, 118250063190, 1792012416075, 28672198657200, 491536207523025, 8847651735414450, 169292834944205175, 3385856698884103500, 71531660838216529125, 1573696538440763640750

Offset: 0

Ilya Gutkovskiy, Dec 19 2017

nonn

			arcsin(x)/(1 - x) = x/1! + 2*x^2/2! + 7*x^3/3! + 28*x^4/4! + 149*x^5/5! + ...

Muhammad Adam Dombrowski and Gregory Dresden, Areas Between Cosines, arXiv:2404.17694 [math.CO], 2024.

Cf. A001147, A001818, A009551, A081358, A281964, A296727.

a:=series(arcsin(x)/(1 - x),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019

nmax = 22; CoefficientList[Series[ArcSin[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[-I Log[I x + Sqrt[1 - x^2]]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

first(n) = x='x+O('x^n); Vec(serlaplace(asin(x)/(1 - x)), -n) \\ Iain Fox, Dec 19 2017

E.g.f.: -i*log(i*x + sqrt(1 - x^2))/(1 - x), where i is the imaginary unit.
a(n) ~ n! * Pi/2. - Vaclav Kotesovec, Dec 20 2017
a(2*n) = 2*n*a(2*n-1). - Greg Dresden, Apr 04 2024
a(2*n+1) = (2*n+1)*(2*n)*a(2*n-1) + ((2*n-1)!!)^2, using the double factorial notation from A001147. - Greg Dresden, Apr 11 2024

A296727 Expansion of e.g.f. arcsinh(x)/(1 - x).

Original entry on oeis.org

0, 1, 2, 5, 20, 109, 654, 4353, 34824, 324441, 3244410, 34795485, 417545820, 5536151685, 77506123590, 1144330385625, 18309286170000, 315366695240625, 5676600514331250, 106667957800963125, 2133359156019262500, 45229212438054868125, 995042673637207098750, 22696937952367956440625

Offset: 0

Ilya Gutkovskiy, Dec 19 2017

nonn

			arcsinh(x)/(1 - x) = x/1! + 2*x^2/2! + 5*x^3/3! + 20*x^4/4! + 109*x^5/5! + ...

Cf. A001818, A009628, A081358, A186763, A281964, A296726.

a:=series(arcsinh(x)/(1 - x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

nmax = 23; CoefficientList[Series[ArcSinh[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[x + Sqrt[1 + x^2]]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

first(n) = x='x+O('x^n); Vec(serlaplace(asinh(x)/(1 - x)), -n) \\ Iain Fox, Dec 19 2017

E.g.f.: log(x + sqrt(1 + x^2))/(1 - x).

a(n) ~ n! * log(1 + sqrt(2)). - Vaclav Kotesovec, Dec 20 2017

A296788 Expansion of e.g.f. exp(x*arcsinh(x)) (even powers only).

Original entry on oeis.org

1, 2, 8, 54, 104, 18810, -1648428, 247726374, -49445941200, 12841169289714, -4206667789245780, 1697448414191239710, -827415782970517712376, 479396168140498731959850, -325673237888367403728512700, 256401822876859593450127851030, -231597610351491427264049084814240

Offset: 0

Ilya Gutkovskiy, Dec 20 2017

sign

			exp(x*arcsinh(x)) = 1 + 2*x^2/2! + 8*x^4/4! + 54*x^6/6! + 104*x^8/8! + ...

Cf. A001818, A006228, A009214, A009233, A079484, A259647, A296787, A296789.

nmax = 16; Table[(CoefficientList[Series[Exp[x ArcSinh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 16; Table[(CoefficientList[Series[(x + Sqrt[1 + x^2])^x, {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*arcsinh(x)).

a(n) ~ -(-1)^n * 2^(2*n) * n^(2*n-1) / exp(2*n + Pi/2). - Vaclav Kotesovec, Dec 21 2017

A003955 a(n) = (2n + 4) (135*...(2n+1))^2.

Original entry on oeis.org

4, 54, 1800, 110250, 10716300, 1512784350, 292183491600, 73958946311250, 23749039426612500, 9430743556307823750, 4537044990907363935000, 2600104866872495148416250, 1750070583471871734510937500, 1366930130733208386919792968750, 1226227455943070136959515612500000

Offset: 0

N. J. A. Sloane, Joe Keane (jgk(AT)jgk.org)

nonn

G. C. Greubel, Table of n, a(n) for n = 0..222

Cf. A001818.

F:=Factorial;; List([0..20], n-> F(n+2)*F(n+1)*Binomial(2*n+2, n+1)^2/2^(2*n+1) ); # G. C. Greubel, Sep 24 2019

F:=Factorial; [F(n+2)*F(n+1)*Binomial(2*n+2,n+1)^2/2^(2*n+1): n in [0..20]]; // G. C. Greubel, Sep 24 2019

seq((n+2)!*(n+1)!*binomial(2*n+2, n+1)^2/2^(2*n+1), n=0..20); # G. C. Greubel, Sep 24 2019

Table[(n+2)!*(n+1)!*Binomial[2*n+2, n+1]^2/2^(2*n+1), {n,0,20}] (* G. C. Greubel, Sep 24 2019 *)

vector(21, n, (n+1)!*n!*binomial(2*n, n)^2/2^(2*n-1) ) \\ G. C. Greubel, Sep 24 2019

f=factorial; [f(n+2)*f(n+1)*binomial(2*n+2,n+1)^2/2^(2*n+1) for n in (0..20)] # G. C. Greubel, Sep 24 2019

Equals (2*n+4) * A001818(n+1).

Equals (n+2)!*(n+1)!*binomial(2*n+2, n+1)^2/2^(2*n+1). - G. C. Greubel, Sep 24 2019

More terms added by G. C. Greubel, Sep 24 2019

A151816 a(n) = (2n)! - ((2n-1)!!)^2.

Original entry on oeis.org

0, 1, 15, 495, 29295, 2735775, 370945575, 68916822975, 16813959537375, 5214921734397375, 2004231846526284375, 934957186489800849375, 520444368391989625959375, 340788940288324502208609375, 259324006920606914270844234375, 226933251813970116856323617109375, 226305693647403205116652558922109375

Offset: 0

N. J. A. Sloane, Jul 03 2009

nonn

This was (incorrectly) proposed as a formula for A001818(2n).

Tewodros Amdeberhan, Adriana Duncan, Victor H. Moll, and Vaishavi Sharma, Filter integrals for orthogonal polynomials, arXiv:2012.05040 [math.CA], 2020.

Cf. A000142, A010050, A001147, A001818.

Bisection of A088979.

seq((2*n)! - doublefactorial(2*n-1)^2, n=0..16); # Georg Fischer, Apr 19 2024

a(n) = A000142(2*n) - A001147(n)^2.

a(n) = A010050(n) - A001818(n).

Definition corrected by Georg Fischer, Apr 18 2024

A204420 Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.

Original entry on oeis.org

1, 0, 1, 0, 6, 3, 0, 120, 90, 15, 0, 5040, 4620, 1260, 105, 0, 362880, 378000, 132300, 18900, 945, 0, 39916800, 45571680, 18711000, 3534300, 311850, 10395, 0, 6227020800, 7628100480, 3511347840, 794593800, 94594500, 5675670, 135135

Offset: 0

José H. Nieto S., Jan 15 2012

The row polynomials t(n,x):= sum(T(n,k)*x^k, k=0..n) satisfy the recurrence relation t(n,x) = (2n-1)*(x+2n-2)*t(n-1,x), with t(0,x)=1, hence t(n,x)=(2n-1)!!*x(x+2)(x+4)...(x+2n-2).

			1;
0,        1,
0,        6,        3;
0,      120,       90,       15;
0,     5040,     4620,     1260,     105;
0,   362880,   378000,   132300,   18900,    945;
0, 39916800, 45571680, 18711000, 3534300, 311850, 10395;

Alois P. Heinz, Rows n = 0..85, flattened
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Angelo Lucia and Amanda Young, A Nonvanishing Spectral Gap for AKLT Models on Generalized Decorated Graphs, arXiv:2212.11872 [math-ph], 2022.

Cf. A049218, A060523, A060524, A132393, A048994.

Row sums give: A001818. - Alois P. Heinz, Jul 21 2013

T_row:= proc(n) local k; seq(doublefactorial(2*n-1)*2^(n-k)* coeff(expand(pochhammer(x, n)), x, k), k=0..n) end: seq(T_row(n), n=0..10);

nn=12;Prepend[Map[Prepend[Select[#,#>0&],0]&,Table[(Range[0, nn]!CoefficientList[ Series[(1-x^2)^(-y/2),{x,0,nn}],{x,y}])[[n]],{n,3,nn,2}]],{1}]//Grid (* Geoffrey Critzer, Jul 21 2013 *)

T(n,k) = (2*n)!/(2^n*n!)*(-2)^(n-k)*stirling(n,k,1); \\ Andrew Howroyd, Feb 12 2018

T(n,k) = (2n-1)!!*2^(n-k)*A132393(n,k).
T(n,k) = (2n-1)T(n-1,k-1) + (2n-1)(2n-2)*T(n-1,k); T(0,0)=1, T(n,0)=0 for n>0,
T(n,n) = (2n-1)!! = A001147(n).
T(n,1) = (2n-1)! = A009445(n-1).

cp's OEIS Frontend

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