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 A062139 Coefficient triangle of generalized Laguerre polynomials n!*L(n,2,x) (rising powers of x). 16
 1, 3, -1, 12, -8, 1, 60, -60, 15, -1, 360, -480, 180, -24, 1, 2520, -4200, 2100, -420, 35, -1, 20160, -40320, 25200, -6720, 840, -48, 1, 181440, -423360, 317520, -105840, 17640, -1512, 63, -1, 1814400, -4838400, 4233600 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The row polynomials s(n,x) := n!*L(n,2,x) = Sum_{m=0..n} a(n,m)*x^m have e.g.f. exp(-z*x/(1-z))/(1-z)^3. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), with polynomials p(n,x) = Sum_{m=1..n} |A008297(n,m)|*(-x)^m, n >= 1 and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference). This unsigned matrix is embedded in the matrix for n!*L(n,-2,-x). Introduce 0,0 to each unsigned row and then add 1,-1,1 to the array as the first two rows to generate n!*L(n,-2,-x). - Tom Copeland, Apr 20 2014 The unsigned n-th row reverse polynomial equals the numerator polynomial of the finite continued fraction 1 - x/(1 + (n+1)*x/(1 + n*x/(1 + n*x/(1 + ... + 2*x/(1 + 2*x/(1 + x/(1 + x/(1)))))))). Cf. A089231. The denominator polynomial of the continued fraction is the (n+1)-th row polynomial of A144084. An example is given below. - Peter Bala, Oct 06 2019 LINKS Indranil Ghosh, Rows 0..125, flattened Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19. Index entries for sequences related to Laguerre polynomials FORMULA T(n, m) = ((-1)^m)*n!*binomial(n+2, n-m)/m!. E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^3), m >= 0. n!*L(n,2,x) = (n+2)!*hypergeom([-n],[3],x)/2. - Peter Luschny, Apr 08 2015 From Werner Schulte, Mar 24 2024: (Start) T(n, k) = (n+k+2) * T(n-1, k) - T(n-1, k-1) with initial values T(0, 0) = 1 and T(i, j) = 0 if j < 0 or j > i. T = T^(-1), i.e., T is matrix inverse of T. (End) EXAMPLE Triangle begins: 1; 3, -1; 12, -8, 1; 60, -60, 15, -1; 360, -480, 180, -24, 1; 2520, -4200, 2100, -420, 35, -1; ... 2!*L(2,2,x) = 12 - 8*x + x^2. Unsigned row 3 polynomial in reverse form as the numerator of a continued fraction: 1 - x/(1 + 4*x/(1 + 3*x/(1 + 3*x/(1 + 2*x/(1 + 2*x/(1 + x/(1 + x))))))) = (60*x^3 + 60*x^2 + 15*x + 1)/(24*x^4 + 96*x^3 + 72*x^2 + 16*x + 1). - Peter Bala, Oct 06 2019 MAPLE with(PolynomialTools): p := n -> (n+2)!*hypergeom([-n], [3], x)/2: seq(CoefficientList(simplify(p(n)), x), n=0..9); # Peter Luschny, Apr 08 2015 MATHEMATICA Flatten[Table[((-1)^m)*n!*Binomial[n+2, n-m]/m!, {n, 0, 8}, {m, 0, n}]] (* Indranil Ghosh, Feb 24 2017 *) PROG (PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1(((-1)^k)*n!*binomial(n+2, n-k)/k!, ", "); ); print(); ); } \\ Michel Marcus, May 06 2014 (PARI) row(n) = Vecrev(n!*pollaguerre(n, 2)); \\ Michel Marcus, Feb 06 2021 (Python) import math f=math.factorial def C(n, r):return f(n)//f(r)//f(n-r) i=0 for n in range(16): for m in range(n+1): i += 1 print(i, ((-1)**m)*f(n)*C(n+2, n-m)//f(m)) # Indranil Ghosh, Feb 24 2017 (Python) from functools import cache @cache def T(n, k): if k < 0 or k > n: return 0 if k == n: return (-1)**n return (n + k + 2) * T(n-1, k) - T(n-1, k-1) for n in range(7): print([T(n, k) for k in range(n + 1)]) # Peter Luschny, Mar 25 2024 CROSSREFS For m=0..5 the (unsigned) columns give A001710, A005990, A005461, A062193-A062195. The row sums (signed) give A062197, the row sums (unsigned) give A052852. Cf. A021009, A062137-A062140, A066667, A089231, A144084. Sequence in context: A049458 A143492 A243662 * A156366 A144353 A356146 Adjacent sequences: A062136 A062137 A062138 * A062140 A062141 A062142 KEYWORD sign,easy,tabl AUTHOR Wolfdieter Lang, Jun 19 2001 STATUS approved

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Last modified September 15 11:45 EDT 2024. Contains 375938 sequences. (Running on oeis4.)