A321965
a(n) = n! [x^n] exp((1/(x - 1)^2 - 1)/2)/(1 - x).
Original entry on oeis.org
1, 2, 8, 46, 338, 2996, 30952, 364148, 4797116, 69854968, 1113018176, 19244304872, 358608737368, 7160626365296, 152458303437728, 3446434090192816, 82412163484132112, 2077739630757428768, 55068742629150564736, 1530394053934299827168, 44490672191650220419616
Offset: 0
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egf := exp((1/(x - 1)^2 - 1)/2)/(1 - x): ser := series(egf, x, 22):
seq(n!*coeff(ser, x, n), n=0..20);
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CoefficientList[Exp[(1/(x - 1)^2 - 1)/2]/(1 - x) + O[x]^21, x] Range[0, 20]! (* Jean-François Alcover, Jan 01 2019 *)
A322944
Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.
Original entry on oeis.org
1, 1, 1, 2, 6, 1, 6, 38, 15, 1, 24, 272, 188, 28, 1, 120, 2200, 2340, 580, 45, 1, 720, 19920, 30280, 11040, 1390, 66, 1, 5040, 199920, 413560, 206920, 37450, 2842, 91, 1, 40320, 2204160, 5989760, 3931200, 955920, 102816, 5208, 120, 1
Offset: 0
Triangle starts:
[0] 1;
[1] 1, 1;
[2] 2, 6, 1;
[3] 6, 38, 15, 1;
[4] 24, 272, 188, 28, 1;
[5] 120, 2200, 2340, 580, 45, 1;
[6] 720, 19920, 30280, 11040, 1390, 66, 1;
[7] 5040, 199920, 413560, 206920, 37450, 2842, 91, 1;
Production matrix starts:
1;
1, 1;
3, 5, 1;
6, 18, 9, 1;
6, 42, 45, 13, 1;
0, 48, 132, 84, 17, 1;
0, 0, 180, 300, 135, 21, 1;
0, 0, 0, 480, 570, 198, 25, 1;
A321966 (m=2), this sequence (m=3).
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P := proc(n) option remember; local a, b, c, d;
a := n -> 4*n-3; b := n -> 3*(n-1)*(2*n-3);
c := n -> (n-1)*(n-2)*(4*n-9); d := n -> (n-2)*(n-1)*(n-3)^2;
if n = 0 then return 1 fi;
if n = 1 then return x + 1 fi;
if n = 2 then return x^2 + 6*x + 2 fi;
if n = 3 then return x^3 + 15*x^2 + 38*x + 6 fi;
expand((x+a(n))*P(n-1) - b(n)*P(n-2) + c(n)*P(n-3) - d(n)*P(n-4)) end:
seq(print(P(n)), n=0..9); # Computes the polynomials.
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a[n_] := 4n - 3;
b[n_] := 3(n - 1)(2n - 3);
c[n_] := (n - 1)(n - 2)(4n - 9);
d[n_] := (n - 2)(n - 1)(n - 3)^2;
P[n_] := P[n] = Switch[n, 0, 1, 1, x + 1, 2, x^2 + 6x + 2, 3, x^3 + 15x^2 + 38x + 6, _, Expand[(x + a[n]) P[n - 1] - b[n] P[n - 2] + c[n] P[n - 3] - d[n] P[n - 4]]];
Table[CoefficientList[P[n], x], {n, 0, 9}] (* Jean-François Alcover, Jun 15 2019, from Maple *)
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# uses[riordan_square from A321620]
R = riordan_square((1 - 3*x)^(-1/3), 9, True).inverse()
for n in (0..8): print([(-1)^(n-k)*c for (k, c) in enumerate(R.row(n)[:n+1])])
Showing 1-2 of 2 results.
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