A166357
Exponential Riordan array [1+x*arctanh(x), x].
Original entry on oeis.org
1, 0, 1, 2, 0, 1, 0, 6, 0, 1, 8, 0, 12, 0, 1, 0, 40, 0, 20, 0, 1, 144, 0, 120, 0, 30, 0, 1, 0, 1008, 0, 280, 0, 42, 0, 1, 5760, 0, 4032, 0, 560, 0, 56, 0, 1, 0, 51840, 0, 12096, 0, 1008, 0, 72, 0, 1, 403200, 0, 259200, 0, 30240, 0, 1680, 0, 90, 0, 1
Offset: 0
Triangle begins
1;
0, 1;
2, 0, 1;
0, 6, 0, 1;
8, 0, 12, 0, 1;
0, 40, 0, 20, 0, 1;
144, 0, 120, 0, 30, 0, 1;
0, 1008, 0, 280, 0, 42, 0, 1;
5760, 0, 4032, 0, 560, 0, 56, 0, 1;
0, 51840, 0, 12096, 0, 1008, 0, 72, 0, 1;
403200, 0, 259200, 0, 30240, 0, 1680, 0, 90, 0, 1;
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(* The function RiordanArray is defined in A256893. *)
RiordanArray[1 + # ArcTanh[#]&, #&, 11, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)
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T(n,k)={binomial(n,k)*(n-k)!*polcoef(1 + x*atanh(x + O(x^max(1, n-k))), n-k)} \\ Andrew Howroyd, Aug 17 2018
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T(n,k)=if(k>=n, n==k, binomial(n, k)*if((n-k)%2, 0, (n-k-1)! + (n-k-2)!)) \\ Andrew Howroyd, Aug 17 2018
A166359
Diagonal sums of the exponential Riordan array [1+x*arctanh(x), x], A166357.
Original entry on oeis.org
1, 3, 15, 197, 6909, 459383, 48252699, 7299708105, 1499523879481, 401147660278507, 135421121289695655, 56285769483090611085, 28237152577765405343285, 16821061018350834178553055
Offset: 0
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Table[1 + Sum[Binomial[n + k, 2*k] * ((2*k-1)! + (2*k-2)!), {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 17 2018 *)
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a(n)={1 + sum(k=1, n, binomial(n + k, 2*k) * ((2*k-1)! + (2*k-2)!))} \\ Andrew Howroyd, Aug 17 2018
A296787
Expansion of e.g.f. exp(x*arctan(x)) (even powers only).
Original entry on oeis.org
1, 2, 4, 24, -496, 36000, -3753408, 556961664, -111591202560, 29054584410624, -9541382573767680, 3858875286730168320, -1884995591107521540096, 1094305223336273239449600, -744771228363250138965196800, 587358379156469629707528929280
Offset: 0
exp(x*arctan(x)) = 1 + 2*x^2/2! + 4*x^4/4! + 24*x^6/6! - 496*x^8/8! + ...
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nmax = 15; Table[(CoefficientList[Series[Exp[x ArcTan[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 15; Table[(CoefficientList[Series[Exp[(I/2) x (Log[1 - I x] - Log[1 + I x])], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A296789
Expansion of e.g.f. exp(x*arctanh(x)) (even powers only).
Original entry on oeis.org
1, 2, 20, 504, 24464, 1959840, 234852672, 39370660224, 8799246209280, 2528787321598464, 908585701684024320, 399070678264750356480, 210373049449102957645824, 131083661069772517440921600, 95304505860052894815543705600, 79961055068441273887848131297280
Offset: 0
exp(x*arctanh(x)) = 1 + 2*x^2/2! + 20*x^4/4! + 504*x^6/6! + 24464*x^8/8! + ...
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nmax = 15; Table[(CoefficientList[Series[Exp[x ArcTanh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 15; Table[(CoefficientList[Series[Exp[x (Log[1 + x] - Log[1 - x])/2], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A166358
Row sums of exponential Riordan array [1+x*arctanh(x), x], A166357.
Original entry on oeis.org
1, 1, 3, 7, 21, 61, 295, 1331, 10409, 65017, 694411, 5454879, 73145149, 689074101, 11090013103, 121652191051, 2282132463953, 28550033871857, 611369381873683, 8587415858721079, 206626962757626981, 3219065122124476717
Offset: 0
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(* The function RiordanArray is defined in A256893. *)
nmax = 21; R = RiordanArray[1 + # ArcTanh[#]&, #&, nmax + 1, True];
Total /@ R (* Jean-François Alcover, Jul 20 2019 *)
Showing 1-5 of 5 results.
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