A001818 -id:A001818 - cp's OEIS frontend

A296979 Expansion of e.g.f. arcsin(log(1 + x)).

0, 1, -1, 3, -12, 68, -480, 4144, -42112, 494360, -6581880, 98079696, -1617373296, 29245459176, -575367843960, 12235339942344, -279650131845120, 6836254328079936, -177979145883651648, 4916243253642325056, -143602294106947553280, 4422411460743707222784

Offset: 0

Ilya Gutkovskiy, Dec 22 2017

sign

			arcsin(log(1 + x)) = x^1/1! - x^2/2! + 3*x^3/3! - 12*x^4/4! + 68*x^5/5! - 480*x^6/6! + ...

Cf. A001710, A001818, A003703, A003708, A009024, A009454, A009775, A104150, A189815, A296980, A296981, A296982.

a:=series(arcsin(log(1+x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019

nmax = 21; CoefficientList[Series[ArcSin[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[-I Log[I Log[1 + x] + Sqrt[1 - Log[1 + x]^2]], {x, 0, nmax}], x] Range[0, nmax]!

a(n) ~ -(-1)^n * n^(n-1) / (exp(1) - 1)^(n - 1/2). - Vaclav Kotesovec, Mar 26 2019

A296980 Expansion of e.g.f. arcsinh(log(1 + x)).

Original entry on oeis.org

0, 1, -1, 1, 0, -2, -30, 446, -3248, 12412, 16020, -211356, -10756944, 284038272, -3556910448, 19122463296, 135073768320, -1286054192304, -108801241372368, 3952903127312016, -65667347037774720, 339816855220730784, 8862271481944986336

Offset: 0

Ilya Gutkovskiy, Dec 22 2017

sign

			arcsinh(log(1 + x)) = x^1/1! - x^2/2! + x^3/3! - 2*x^5/5! - 30*x^6/6! + ...

Cf. A001710, A001818, A003703, A003708, A009024, A009454, A009775, A104150, A296435, A296979, A296981, A296982.

a:=series(arcsinh(log(1+x)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[ArcSinh[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Log[Log[1 + x] + Sqrt[1 + Log[1 + x]^2]], {x, 0, nmax}], x] Range[0, nmax]!

A301942 Expansion of e.g.f. arcsin(x)/cos(x) (odd powers only).

Original entry on oeis.org

1, 4, 44, 1016, 42384, 2908544, 306305856, 46659144832, 9760451385600, 2683733034474496, 936308392553036800, 403127865773461755904, 209562975305232836300800, 129255511221696545852424192, 93252273300325219683758915584, 77766048645578119241905858314240

Offset: 0

Ilya Gutkovskiy, Apr 09 2018

nonn

			arcsin(x)/cos(x) = x/1! + 4*x^3/3! + 44*x^5/5! + 1016*x^7/7! + 42384*x^9/9! + ...

Cf. A000182, A000364, A000795, A001818, A002084, A003701, A003702, A012782, A296741, A302444, A302542, A302543.

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

a(n) = (2*n+1)! * [x^(2*n+1)] arcsin(x)/cos(x).

A302444 Expansion of e.g.f. arcsinh(x)/cos(x) (odd powers only).

Original entry on oeis.org

1, 2, 24, 216, 15936, -77056, 90991744, -8523712768, 2731708067840, -684815907467264, 268028469798256640, -114888252320482000896, 62022733722259702579200, -38635369828053720937463808, 28349537098304682205749968896, -23874826868622028919177351004160

Offset: 0

Ilya Gutkovskiy, Apr 09 2018

sign

			arcsinh(x)/cos(x) = x/1! + 2*x^3/3! + 24*x^5/5! + 216*x^7/7! + 15936*x^9/9! - 77056*x^11/11! + ...

Cf. A000182, A000364, A000795, A001818, A002084, A003701, A003702, A012821, A296742, A301942, A302542, A302543.

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

a(n) = (2*n+1)! * [x^(2*n+1)] arcsinh(x)/cos(x).

A302606 a(n) = n! * [x^n] exp(nx)arcsinh(x).

Original entry on oeis.org

0, 1, 4, 26, 240, 2884, 42660, 748544, 15185856, 349574544, 9000902500, 256293989984, 7996078704240, 271246034903232, 9939835626507332, 391303051339622400, 16469438021801262848, 737992773619777599744, 35077254665501330210628, 1762671472887447792620032

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A001818, A002866, A291483, A302583, A302584, A302585, A302586, A302587, A302605, A302608, A302609.

Table[n! SeriesCoefficient[Exp[n x] ArcSinh[x], {x, 0, n}], {n, 0, 19}]

a(n) ~ arcsinh(1) * n^n = log(1 + sqrt(2)) * n^n. - Vaclav Kotesovec, Jun 09 2019

a(n) = Sum_{k=1..n, k odd} (-1)^((k-1)/2)*binomial(n,k)*(k-2)!!^2*n^(n-k). - Fabian Pereyra, Oct 05 2024

A325221 E.g.f.: C(x,k) = cn( i * Integral C(x,k) dx, k), where C(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2n)k^(2j)/(2n)!, as a triangle of coefficients T(n,j) read by rows.

Original entry on oeis.org

1, 1, 0, 5, 4, 0, 61, 148, 16, 0, 1385, 6744, 2832, 64, 0, 50521, 410456, 383856, 47936, 256, 0, 2702765, 32947964, 54480944, 17142784, 780544, 1024, 0, 199360981, 3402510924, 8760740640, 5199585280, 686711040, 12555264, 4096, 0, 19391512145, 441239943664, 1632067372896, 1569971730560, 419867864320, 26090711040, 201199616, 16384, 0, 2404879675441, 70347660061552, 353538702361888, 502094919789184, 227204970315520, 30892394850304, 965223559168, 3220652032, 65536, 0

Offset: 0

Paul D. Hanna, Apr 13 2019

Equals a row reversal of triangle A322232.

Compare to cn(x,k) = 1 - Integral sn(x,k)*dn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions (see triangle A060627).

			E.g.f.: C(x,k) = 1 + x^2/2! + (5 + 4*k^2)*x^4/4! + (61 + 148*k^2 + 16*k^4)*x^6/6! + (1385 + 6744*k^2 + 2832*k^4 + 64*k^6)*x^8/8! + (50521 + 410456*k^2 + 383856*k^4 + 47936*k^6 + 256*k^8)*x^10/10! + (2702765 + 32947964*k^2 + 54480944*k^4 + 17142784*k^6 + 780544*k^8 + 1024*k^10)*x^12/12! + (199360981 + 3402510924*k^2 + 8760740640*k^4 + 5199585280*k^6 + 686711040*k^8 + 12555264*k^10 + 4096*k^12)*x^14/14! + ...
such that C(x,k) = cn( i * Integral C(x,k) dx, k).
This triangle of coefficients T(n,j) of x^(2*n)*k^(2*j)/(2*n)! in e.g.f. C(x,k) begins:
1;
1, 0;
5, 4, 0;
61, 148, 16, 0;
1385, 6744, 2832, 64, 0;
50521, 410456, 383856, 47936, 256, 0;
2702765, 32947964, 54480944, 17142784, 780544, 1024, 0;
199360981, 3402510924, 8760740640, 5199585280, 686711040, 12555264, 4096, 0;
19391512145, 441239943664, 1632067372896, 1569971730560, 419867864320, 26090711040, 201199616, 16384, 0;
2404879675441, 70347660061552, 353538702361888, 502094919789184, 227204970315520, 30892394850304, 965223559168, 3220652032, 65536, 0; ...
RELATED SERIES.
The related series S(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
S(x,k) = x + (2 + 1*k^2)*x^3/3! + (16 + 28*k^2 + 1*k^4)*x^5/5! + (272 + 1032*k^2 + 270*k^4 + 1*k^6)*x^7/7! + (7936 + 52736*k^2 + 36096*k^4 + 2456*k^6 + 1*k^8)*x^9/9! + (353792 + 3646208*k^2 + 4766048*k^4 + 1035088*k^6 + 22138*k^8 + 1*k^10)*x^11/11! + (22368256 + 330545664*k^2 + 704357760*k^4 + 319830400*k^6 + 27426960*k^8 + 199284*k^10 + 1*k^12)*x^13/13! + (1903757312 + 38188155904*k^2 + 120536980224*k^4 + 93989648000*k^6 + 18598875760*k^8 + 702812568*k^10 + 1793606*k^12 + 1*k^14)*x^15/15! + ...
The related series D(x,k), where D(x,k)^2 - k^2*S(x,k)^2 = 1, starts
D(x,k) = 1 + k^2*x^2/2! + (8*k^2 + 1*k^4)*x^4/4! + (136*k^2 + 88*k^4 + 1*k^6)*x^6/6! + (3968*k^2 + 6240*k^4 + 816*k^6 + 1*k^8)*x^8/8! + (176896*k^2 + 513536*k^4 + 195216*k^6 + 7376*k^8 + 1*k^10)*x^10/10! + (11184128*k^2 + 51880064*k^4 + 39572864*k^6 + 5352544*k^8 + 66424*k^10 + 1*k^12)*x^12/12! + (951878656*k^2 + 6453433344*k^4 + 8258202240*k^6 + 2458228480*k^8 + 139127640*k^10 + 597864*k^12 + 1*k^14)*x^14/14! + ...

Cf. A325220 (S), A325222(D).

Cf. A322232 (row reversal).

N=10;
{S=x; C=1; D=1; for(i=1, 2*N, S = intformal(C^2*D +O(x^(2*N+1))); C = 1 + intformal(S*C*D); D = 1 + k^2*intformal(S*C^2)); }
{T(n,j) = (2*n)!*polcoeff(polcoeff(C, 2*n, x), 2*j, k)}
for(n=0, N, for(j=0, n, print1( T(n,j), ", ")) ; print(""))

E.g.f. C = C(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n)*k^(2*j)/(2*n)!, along with related series S = S(x,k) and D = D(x,k), satisfies:
(1a) S = Integral C^2*D dx.
(1b) C = 1 + Integral S*C*D dx.
(1c) D = 1 + k^2 * Integral S*C^2 dx.
(2a) C^2 - S^2 = 1.
(2b) D^2 - k^2*S^2 = 1.
(3a) C + S = exp( Integral C*D dx ).
(3b) D + k*S = exp( k * Integral C^2 dx ).
(4a) S = sinh( Integral C*D dx ).
(4b) S = sinh( k * Integral C^2 dx ) / k.
(4c) C = cosh( Integral C*D dx ).
(4d) D = cosh( k * Integral C^2 dx ).
(5a) d/dx S = C^2*D.
(5b) d/dx C = S*C*D.
(5c) d/dx D = k^2 * S*C^2.
Given sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions, with i^2 = -1, k' = sqrt(1-k^2), then
(6a) S = -i * sn( i * Integral C dx, k),
(6b) C = cn( i * Integral C dx, k),
(6c) D = dn( i * Integral C dx, k).
(7a) S = sc( Integral C dx, k') = sn(Integral C dx, k')/cn(Integral C dx, k'),
(7b) C = nc( Integral C dx, k') = 1/cn(Integral C dx, k'),
(7c) D = dc( Integral C dx, k') = dn(Integral C dx, k')/cn(Integral C dx, k').
Row sums equal ( (2*n)!/(n!*2^n) )^2 = A001818(n), the squares of the odd double factorials.
Column T(n,0) = A000364(n), for n>=0, where A000364 is the secant numbers.

A325222 E.g.f.: D(x,k) = dn( i * Integral C(x,k) dx, k) such that C(x,k) = cn( i * Integral C(x,k) dx, k), where D(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2n)k^(2j)/(2n)!, as a triangle of coefficients T(n,j) read by rows.

Original entry on oeis.org

1, 0, 1, 0, 8, 1, 0, 136, 88, 1, 0, 3968, 6240, 816, 1, 0, 176896, 513536, 195216, 7376, 1, 0, 11184128, 51880064, 39572864, 5352544, 66424, 1, 0, 951878656, 6453433344, 8258202240, 2458228480, 139127640, 597864, 1, 0, 104932671488, 978593947648, 1889844670464, 994697838080, 137220256000, 3535586112, 5380832, 1, 0, 14544442556416, 178568645312512, 485265505927168, 398800479698944, 102950036177920, 7233820923904, 88992306208, 48427552, 1

Offset: 0

Paul D. Hanna, Apr 13 2019

Equals a row reversal of triangle A322231.

Compare to dn(x,k) = 1 - k^2 * Integral sn(x,k)*cn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions.

			E.g.f.: D(x,k) = 1 + k^2*x^2/2! + (8*k^2 + 1*k^4)*x^4/4! + (136*k^2 + 88*k^4 + 1*k^6)*x^6/6! + (3968*k^2 + 6240*k^4 + 816*k^6 + 1*k^8)*x^8/8! + (176896*k^2 + 513536*k^4 + 195216*k^6 + 7376*k^8 + 1*k^10)*x^10/10! + (11184128*k^2 + 51880064*k^4 + 39572864*k^6 + 5352544*k^8 + 66424*k^10 + 1*k^12)*x^12/12! + (951878656*k^2 + 6453433344*k^4 + 8258202240*k^6 + 2458228480*k^8 + 139127640*k^10 + 597864*k^12 + 1*k^14)*x^14/14! + ...
such that D(x,k) = dn( i * Integral C(x,k) dx, k) where C(x,k) = cn( i * Integral C(x,k) dx, k).
This triangle of coefficients T(n,j) of x^(2*n)*k^(2*j)/(2*n)! in e.g.f. D(x,k) begins:
1;
0, 1;
0, 8, 1;
0, 136, 88, 1;
0, 3968, 6240, 816, 1;
0, 176896, 513536, 195216, 7376, 1;
0, 11184128, 51880064, 39572864, 5352544, 66424, 1;
0, 951878656, 6453433344, 8258202240, 2458228480, 139127640, 597864, 1;
0, 104932671488, 978593947648, 1889844670464, 994697838080, 137220256000, 3535586112, 5380832, 1;
0, 14544442556416, 178568645312512, 485265505927168, 398800479698944, 102950036177920, 7233820923904, 88992306208, 48427552, 1; ...
RELATED SERIES.
The related series S(x,k), where D(x,k)^2 - k^2*S(x,k)^2 = 1, starts
S(x,k) = x + (2 + 1*k^2)*x^3/3! + (16 + 28*k^2 + 1*k^4)*x^5/5! + (272 + 1032*k^2 + 270*k^4 + 1*k^6)*x^7/7! + (7936 + 52736*k^2 + 36096*k^4 + 2456*k^6 + 1*k^8)*x^9/9! + (353792 + 3646208*k^2 + 4766048*k^4 + 1035088*k^6 + 22138*k^8 + 1*k^10)*x^11/11! + (22368256 + 330545664*k^2 + 704357760*k^4 + 319830400*k^6 + 27426960*k^8 + 199284*k^10 + 1*k^12)*x^13/13! + (1903757312 + 38188155904*k^2 + 120536980224*k^4 + 93989648000*k^6 + 18598875760*k^8 + 702812568*k^10 + 1793606*k^12 + 1*k^14)*x^15/15! + ...
The related series C(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
C(x,k) = 1 + x^2/2! + (5 + 4*k^2)*x^4/4! + (61 + 148*k^2 + 16*k^4)*x^6/6! + (1385 + 6744*k^2 + 2832*k^4 + 64*k^6)*x^8/8! + (50521 + 410456*k^2 + 383856*k^4 + 47936*k^6 + 256*k^8)*x^10/10! + (2702765 + 32947964*k^2 + 54480944*k^4 + 17142784*k^6 + 780544*k^8 + 1024*k^10)*x^12/12! + (199360981 + 3402510924*k^2 + 8760740640*k^4 + 5199585280*k^6 + 686711040*k^8 + 12555264*k^10 + 4096*k^12)*x^14/14! + ...
which also satisfies C(x,k) = cn( i * Integral C(x,k) dx, k).

Cf. A325220 (S), A325221(C).

Cf. A322231 (row reversal).

N=10;
{S=x; C=1; D=1; for(i=1, 2*N, S = intformal(C^2*D +O(x^(2*N+1))); C = 1 + intformal(S*C*D); D = 1 + k^2*intformal(S*C^2)); }
{T(n,j) = (2*n)!*polcoeff(polcoeff(D, 2*n, x), 2*j, k)}
for(n=0, N, for(j=0, n, print1( T(n,j), ", ")) ; print(""))

E.g.f. D = D(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n)*k^(2*j)/(2*n)!, along with related series S = S(x,k) and C = C(x,k), satisfies:
(1a) S = Integral C^2*D dx.
(1b) C = 1 + Integral S*C*D dx.
(1c) D = 1 + k^2 * Integral S*C^2 dx.
(2a) C^2 - S^2 = 1.
(2b) D^2 - k^2*S^2 = 1.
(3a) C + S = exp( Integral C*D dx ).
(3b) D + k*S = exp( k * Integral C^2 dx ).
(4a) S = sinh( Integral C*D dx ).
(4b) S = sinh( k * Integral C^2 dx ) / k.
(4c) C = cosh( Integral C*D dx ).
(4d) D = cosh( k * Integral C^2 dx ).
(5a) d/dx S = C^2*D.
(5b) d/dx C = S*C*D.
(5c) d/dx D = k^2 * S*C^2.
Given sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions, with i^2 = -1, k' = sqrt(1-k^2), then
(6a) S = -i * sn( i * Integral C dx, k),
(6b) C = cn( i * Integral C dx, k),
(6c) D = dn( i * Integral C dx, k).
(7a) S = sc( Integral C dx, k') = sn(Integral C dx, k')/cn(Integral C dx, k'),
(7b) C = nc( Integral C dx, k') = 1/cn(Integral C dx, k'),
(7c) D = dc( Integral C dx, k') = dn(Integral C dx, k')/cn(Integral C dx, k').
Row sums equal ( (2*n)!/(n!*2^n) )^2 = A001818(n), the squares of the odd double factorials.
Column T(n,n+1) = 2^n*A002105(n+1), for n>=0, where A002105 gives the reduced tangent numbers.

A353972 Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arcsin(x).

Original entry on oeis.org

1, 0, 1, -4, 29, -124, 1583, -17088, 124553, -1152816, 20127867, -262838016, 3978820221, -48595514304, 914656587063, -24441484099584, 370244721585681, -5884988565575424, 162968423791332339, -3855257807841017856, 82014901819948738629, -1934570487417807744000, 58311771938510122952559

Offset: 1

Ilya Gutkovskiy, May 22 2022

sign

Cf. A001818, A067856, A353818, A353913, A354115, A354171, A354274, A354275, A354276.

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = Mod[n, 2] (n - 2)!!/(n (n - 1)!!) - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]

E.g.f.: Sum_{k>=1} A067856(k) * log(1 + arcsin(x^k)) / k.

A354274 Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arcsinh(x).

Original entry on oeis.org

1, 0, -1, 4, -11, -4, -547, 7680, -7751, 81744, -3258663, -9474816, -390445563, 233029824, -964154427, 4193551958016, -18431412645519, 71090090006784, -6436900596281679, 17349989459410944, 834261829219880829, -241960391975347200, -1149793471388581053219

Offset: 1

Ilya Gutkovskiy, May 22 2022

sign

Cf. A001818, A067856, A353819, A353914, A353972, A354116, A354172, A354275, A354276.

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[c[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; c[n_] := c[n] = {1, 0, -1, 0}[[Mod[n, 4, 1]]] (n - 2)!!/(n (n - 1)!!) - b[n, n - 1]; a[n_] := n! c[n]; Table[a[n], {n, 1, 23}]

E.g.f.: Sum_{k>=1} A067856(k) * log(1 + arcsinh(x^k)) / k.

A054479 Number of sets of cycle graphs of 2n nodes where the 2-colored edges alternate colors.

Original entry on oeis.org

1, 0, 6, 120, 6300, 514080, 62785800, 10676746080, 2413521910800, 700039083744000, 253445583029839200, 112033456760809584000, 59382041886244720843200, 37175286835046004765120000, 27139206193305890195912400000, 22852066417535931447551359680000

Offset: 0

Christian G. Bower, Mar 29 2000

nonn

Also number of permutations in the symmetric group S_2n in which cycle lengths are even and greater than 2, cf. A130915. - Vladeta Jovovic, Aug 25 2007

a(n) is also the number of ordered pairs of disjoint perfect matchings in the complete graph on 2n vertices. The sequence A006712 is the number of ordered triples of perfect matchings. - Matt Larson, Jul 23 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001147, A001818, A053871, A006712.

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-2*j)*binomial(n-1, 2*j-1)*(2*j-1)!, j=2..n/2))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..15);  # Alois P. Heinz, Mar 06 2023

Table[(n-1)*(2*n)!^2 * HypergeometricPFQ[{2-n},{3/2-n},-1/2] / (4^n*(n-1/2)*(n!)^2), {n, 0, 20}] (* Vaclav Kotesovec, Mar 29 2014 after Mark van Hoeij *)

x='x+O('x^66); v=Vec(serlaplace(1/(sqrt(exp(x^2)*(1-x^2))))); vector(#v\2,n,v[2*n-1]) \\ Joerg Arndt, May 13 2013

If b(2n)=a(n) then e.g.f. of b is 1/(sqrt(exp(x^2)*(1-x^2))).
a(n) = 4^n*(n-1)*gamma(n+1/2)^2*hypergeom([2-n],[3/2-n],-1/2)/(Pi*(n-1/2)). - Mark van Hoeij, May 13 2013
a(n) ~ 2^(2*n+1) * n^(2*n) / exp(2*n+1/2). - Vaclav Kotesovec, Mar 29 2014

cp's OEIS Frontend

A296979 Expansion of e.g.f. arcsin(log(1 + x)).

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A296980 Expansion of e.g.f. arcsinh(log(1 + x)).

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A301942 Expansion of e.g.f. arcsin(x)/cos(x) (odd powers only).

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A302444 Expansion of e.g.f. arcsinh(x)/cos(x) (odd powers only).

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A302606 a(n) = n! * [x^n] exp(n*x)*arcsinh(x).

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A325221 E.g.f.: C(x,k) = cn( i * Integral C(x,k) dx, k), where C(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n)*k^(2*j)/(2*n)!, as a triangle of coefficients T(n,j) read by rows.

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A325222 E.g.f.: D(x,k) = dn( i * Integral C(x,k) dx, k) such that C(x,k) = cn( i * Integral C(x,k) dx, k), where D(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n)*k^(2*j)/(2*n)!, as a triangle of coefficients T(n,j) read by rows.

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A353972 Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arcsin(x).

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A354274 Product_{n>=1} (1 + x^n)^(a(n)/n!) = 1 + arcsinh(x).

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A302606 a(n) = n! * [x^n] exp(nx)arcsinh(x).

A325221 E.g.f.: C(x,k) = cn( i * Integral C(x,k) dx, k), where C(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2n)k^(2j)/(2n)!, as a triangle of coefficients T(n,j) read by rows.

A325222 E.g.f.: D(x,k) = dn( i * Integral C(x,k) dx, k) such that C(x,k) = cn( i * Integral C(x,k) dx, k), where D(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2n)k^(2j)/(2n)!, as a triangle of coefficients T(n,j) read by rows.