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 A001821 Central factorial numbers. (Formerly M5215 N2269) 2
 1, 30, 1023, 44473, 2475473, 173721912, 15088541896, 1593719752240, 201529405816816, 30092049283982400, 5242380158902146624, 1054368810603158319360, 242558905724502235934976, 63305390270900389045395456, 18607799824329123330114576384 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n-3) is the coefficient of x^4 in Product_{k=0..n} (x + k^2). - Ralf Stephan, Aug 22 2004 REFERENCES J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..100 Takao Komatsu, Convolution identities of poly-Cauchy numbers with level 2, arXiv:2003.12926 [math.NT], 2020. Mircea Merca, A Special Case of the Generalized Girard-Waring Formula J. Integer Sequences, Vol. 15 (2012), Article 12.5.7. FORMULA a(n) = s(n+4,4)^2 - 2*s(n+4,1)*s(n+4,7) + 2*s(n+4,2)*s(n+4,6) - 2*s(n+4,3)*s(n+4,5), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 03 2012 a(n) = 2*(2*n^2 + 6*n + 7)*a(n-1) - 3*(2*n^4 + 8*n^3 + 16*n^2 + 16*n + 7)*a(n-2) + (2*n^2 + 2*n + 1)*(2*n^4 + 4*n^3 + 6*n^2 + 4*n + 1)*a(n-3) - n^8*a(n-4). - Vaclav Kotesovec, Feb 23 2015 a(n) ~ Pi^7 * n^(2*n+7) / (2520 * exp(2*n)). - Vaclav Kotesovec, Feb 23 2015 MAPLE seq(Stirling1(n+4, 4)^2-2*Stirling1(n+4, 1)*Stirling1(n+4, 7)+2*Stirling1(n+4, 2)*Stirling1(n+4, 6) -2*Stirling1(n+4, 3)*Stirling1(n+4, 5), n=0..20); # Mircea Merca, Apr 03 2012 MATHEMATICA Table[StirlingS1[n+4, 4]^2 - 2*StirlingS1[n+4, 1]*StirlingS1[n+4, 7] + 2*StirlingS1[n+4, 2]*StirlingS1[n+4, 6] - 2*StirlingS1[n+4, 3]*StirlingS1[n+4, 5], {n, 0, 20}] (* T. D. Noe, Aug 10 2012 *) PROG (Python) from sympy.functions.combinatorial.numbers import stirling def s(n, k): return stirling(n, k, kind=1) def a(n): return s(n+4, 4)**2 - 2*s(n+4, 1)*s(n+4, 7) + 2*s(n+4, 2)*s(n+4, 6) - 2*s(n+4, 3)*s(n+4, 5) print([a(n) for n in range(15)]) # Michael S. Branicky, Jan 30 2021 CROSSREFS Fourth right-hand column of triangle A008955. Sequence in context: A180812 A292002 A160313 * A027488 A269541 A280216 Adjacent sequences:  A001818 A001819 A001820 * A001822 A001823 A001824 KEYWORD nonn AUTHOR EXTENSIONS More terms from Ralf Stephan, Aug 22 2004 STATUS approved

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Last modified December 6 12:38 EST 2021. Contains 349563 sequences. (Running on oeis4.)