A381430
E.g.f. A(x) satisfies A(x) = 1 + sinh(x*A(x)^3).
Original entry on oeis.org
1, 1, 6, 73, 1368, 34861, 1126368, 44135701, 2034072960, 107823563641, 6463383851520, 432331180935457, 31924171503581184, 2579483385868484005, 226383845487041421312, 21445302563389991287981, 2180974075392495296544768, 237009522316557393020262001, 27409082977094100068471537664
Offset: 0
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a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(3*n+1, k)*a136630(n, k))/(3*n+1);
A381518
Expansion of e.g.f. ( (1/x) * Series_Reversion( x/(1 + sin(x))^2 ) )^(1/2).
Original entry on oeis.org
1, 1, 4, 29, 304, 4141, 68832, 1337881, 29432576, 712263961, 18403873280, 487814777141, 12296236382208, 230142147098501, -2906327530115072, -800177574047914831, -75835523291585773568, -6054072134316123116495, -459749417224473755910144, -34556942957229166465685555
Offset: 0
-
a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+1, k)*I^(n-k)*a136630(n, k))/(2*n+1);
A107404
Expansion of e.g.f. 1/(1 - sinh(x))^2.
Original entry on oeis.org
1, 2, 6, 26, 144, 962, 7536, 67706, 685824, 7730882, 95970816, 1300815386, 19113775104, 302616787202, 5135568746496, 92996021795066, 1789758460329984, 36479831022049922, 785020114093080576, 17785273588395966746, 423150055005134782464, 10548427254444904799042
Offset: 0
-
E(x):=1/(1-sinh(x))^2: f[0]:=E(x): for n from 1 to 30 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..30);
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CoefficientList[Series[1/(1-Sinh[x])^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)
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a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (k+1)!*a136630(n, k)); \\ Seiichi Manyama, Feb 17 2025
A381262
Expansion of e.g.f. exp( -LambertW(-2 * sinh(x)) / 2 ).
Original entry on oeis.org
1, 1, 5, 50, 749, 15132, 385953, 11907520, 431376345, 17954558928, 844397935517, 44287052219104, 2563077440429701, 162259043437047104, 11154216390820950585, 827464985582299977728, 65889383717510410496689, 5605511011776107945980160, 507429545895353798767136181
Offset: 0
-
a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (2*k+1)^(k-1)*a136630(n, k));
A381263
Expansion of e.g.f. exp( -LambertW(-2 * sin(x)) / 2 ).
Original entry on oeis.org
1, 1, 5, 48, 709, 14152, 356793, 10882648, 389790889, 16040853568, 745908722477, 38681745244032, 2213527304014189, 138556837227204736, 9417928265797994145, 690818806495197538816, 54391227913053881634001, 4575388875753714015748096, 409532433006878699321370197
Offset: 0
-
a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (2*k+1)^(k-1)*I^(n-k)*a136630(n, k));
A381303
E.g.f. A(x) satisfies A(x) = 1/( 1 - sinh(x * A(x)^(1/2)) / A(x)^(3/2) ).
Original entry on oeis.org
1, 1, 0, 1, 4, 1, 32, 183, 192, 4921, 33664, 88573, 2100224, 16487745, 83890176, 1920800731, 17243373568, 143156073841, 3236025171968, 33490813489497, 401094916964352, 9092346624868321, 109434837281013760, 1724106500663768191, 39706910863092875264
Offset: 0
-
a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n/2-k/2+1, k)/(n/2-k/2+1)*a136630(n, k));
A381304
E.g.f. A(x) satisfies A(x) = 1/( 1 - sinh(x * A(x)^(1/2)) / A(x)^(1/2) ).
Original entry on oeis.org
1, 1, 2, 7, 36, 241, 1984, 19461, 222080, 2892361, 42350976, 688911763, 12329035264, 240789209025, 5096898326528, 116247332597833, 2842225449025536, 74165478671163601, 2057366115038003200, 60461340544432547391, 1876511245926278365184, 61336532673286072390321
Offset: 0
-
a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n/2+k/2+1, k)/(n/2+k/2+1)*a136630(n, k));
A381305
E.g.f. A(x) satisfies A(x) = 1/( 1 - sin(x * A(x)^(1/2)) / A(x)^(3/2) ).
Original entry on oeis.org
1, 1, 0, -1, -4, 1, 32, 181, -192, -4919, -31616, 88571, 2089984, 13830545, -83841024, -1884928471, -11874992128, 142704083281, 3085703610368, 16806597846295, -397246640947200, -8257973126103359, -32717082633175040, 1686557057210338589, 33490001971564773376
Offset: 0
-
a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n/2-k/2+1, k)/(n/2-k/2+1)*I^(n-k)*a136630(n, k));
A381306
E.g.f. A(x) satisfies A(x) = 1/( 1 - sin(x * A(x)^(1/2)) / A(x)^(1/2) ).
Original entry on oeis.org
1, 1, 2, 5, 12, 1, -416, -5741, -60800, -543719, -3479424, 6260561, 822338048, 20933340065, 393396789248, 5840683299431, 54344509046784, -481407806103119, -44548560374988800, -1564969488082711811, -40856692743724335104, -812774967576805701599, -8614414458975040831488
Offset: 0
-
a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n/2+k/2+1, k)/(n/2+k/2+1)*I^(n-k)*a136630(n, k));
A381309
E.g.f. A(x) satisfies A(x) = exp( sinh(x * A(x)^(1/2)) / A(x)^(1/2) ).
Original entry on oeis.org
1, 1, 1, 2, 9, 42, 209, 1381, 11121, 96744, 936337, 10323865, 125245457, 1640739336, 23339285601, 359236548033, 5918755368865, 103922094286976, 1941594484205793, 38448924176712705, 803753373207738337, 17693469280066921736, 409266060724837855057, 9922356658347766201841
Offset: 0
-
a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (n/2-k/2+1)^(k-1)*a136630(n, k));