A060926
Row sums of triangle A060923 (even part of bisection of Lucas triangle).
Original entry on oeis.org
1, 5, 29, 149, 765, 3941, 20301, 104565, 538589, 2774149, 14289005, 73599381, 379093501, 1952623525, 10057515149, 51803949749, 266830242845, 1374381274821, 7079122173101, 36462931836885
Offset: 0
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I:=[1,5,29]; [n le 3 select I[n] else 5*Self(n-1) + 4*Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 09 2021
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LinearRecurrence[{5,0,4}, {1,5,29}, 31] (* G. C. Greubel, Apr 09 2021 *)
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def A060926_list(prec):
P. = PowerSeriesRing(QQ, prec)
return P( (1+4*x^2)/(1-5*x-4*x^3) ).list()
A060926_list(30) # G. C. Greubel, Apr 09 2021
A060927
Row sums of triangle A060924 (odd part of bisection of Lucas triangle).
Original entry on oeis.org
3, 13, 65, 337, 1737, 8945, 46073, 237313, 1222345, 6296017, 32429337, 167036065, 860364393, 4431539313, 22825840825, 117570661697, 605579465737, 3119200691985, 16066286106713, 82753748396513
Offset: 0
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R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (3-2*x)/(1-5*x-4*x^3) )); // G. C. Greubel, Apr 07 2021
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CoefficientList[Series[(3-2*x)/(1-5*x-4*x^3), {x, 0, 30}], x] (* G. C. Greubel, Apr 07 2021 *)
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def A060927_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( (3-2*x)/(1-5*x-4*x^3) ).list()
A060927_list(30) # G. C. Greubel, Apr 07 2021
A032428
Coefficients of Jacobi elliptic function c(5,m).
Original entry on oeis.org
1, 99642, 116294673, 47152124264, 11966116940238, 2347836365864484, 393938089395885894, 59752013018382750024, 8470841585571575617239, 1146456994425541774291534, 150221961163114696686151695, 19239380962379456298762250416, 2424371762015227695363084225932
Offset: 5
A032427
Coefficients of Jacobi elliptic function c(4,m).
Original entry on oeis.org
1, 11069, 4494351, 834687179, 109645021894, 11966116940238, 1171517154238290, 107266611330420090, 9412382749388124015, 803475280086029066515, 67362921649153881472361, 5581153512072331417781229
Offset: 0
- A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.
- S. Wrigge, Calculation of the Taylor series expansion coefficients of the Jacobian elliptic function sn(x, k), Math. Comp. 36 (1981), 555-564. [From Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010]
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a:=proc(n) options remember: local m: if n>2 then if n mod 2 = 0 then m:=n/2-1: RETURN(-4*(1+k^2)*a(n-2)+6*k^2*add(binomial(n-2,2*v)*a(2*v)*a(n-2-2*v),v=1..m-1)) else m:=(n-1)/2-1: RETURN(-(1+k^2)*a(n-2)+2*k^2*add(binomial(n-2,2*v+1)*a(2*v+1)*a(n-3-2*v),v=0..m-1)) fi else RETURN([1,2][n]) fi:end: seq(abs(coeff(a(2*i+1),k,8)),i=4..23); # Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010
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a[n_] := a[n] = Module[{m}, If[n > 2, If[Mod[n, 2] == 0, m = n/2 - 1; Return[-4*(1 + k^2)*a[n - 2] + 6*k^2*Sum[Binomial[n - 2, 2*v]*a[2*v]*a[n - 2 - 2*v], {v, 1, m - 1}]], m = (n - 1)/2 - 1; Return[-(1 + k^2)*a[n - 2] + 2*k^2*Sum[Binomial[n - 2, 2*v + 1]*a[2*v + 1]*a[n - 3 - 2*v], {v, 0, m - 1}]]], Return[{1, 2}[[n]]]]];
Table[Abs[Coefficient[a[2*i + 1], k, 8]], {i, 4, 15}] (* Jean-François Alcover, Jul 08 2022, after Herman Jamke's Maple code *)
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010
A032429
Coefficients of Jacobi elliptic function c(6,m).
Original entry on oeis.org
1, 896803, 2949965020, 2504055894564, 1171517154238290, 393938089395885894, 107947764316226205276, 25835579116799316507780, 5632500127524872577252027, 1149330973559307337432235521, 223559382769795167319093086664, 41982964485265754951017173213880
Offset: 6
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