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 A048786 Triangle of coefficients of certain exponential convolution polynomials. 4
 1, 8, 1, 96, 24, 1, 1536, 576, 48, 1, 30720, 15360, 1920, 80, 1, 737280, 460800, 76800, 4800, 120, 1, 20643840, 15482880, 3225600, 268800, 10080, 168, 1, 660602880, 578027520, 144506880, 15052800, 752640, 18816, 224, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS i) p(n,x) := sum(a(n,m)*x^m,m=1..n), p(0,x) := 1, are monic polynomials satisfying p(n,x+y)= sum(binomial(n,k)*p(k,x)*p(n-k,y),k=0..n), (exponential convolution polynomials). ii) In the terminology of the umbral calculus (see reference) p(n,x) are called associated to f(t)= t/(1+4*t). iii) a(n,1)= A034177(n). Also the Bell transform of A034177. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016 Also the fourth power of the unsigned Lah triangular matrix A105278. - Shuhei Tsujie, May 18 2019 Also the number of k-dimensional flats of the extended Shi arrangement of dimension n consisting of hyperplanes x_i - x_j = d (1 <= i < j <= n, -3 <= d <= 4). - Shuhei Tsujie, May 18 2019 REFERENCES S. Roman, The Umbral Calculus, Academic Press, New York, 1984 LINKS N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019. FORMULA a(n, m) = n!*4^(n-m)*binomial(n-1, m-1)/m!, n >= m >= 1; a(n, m) := 0, m>n; a(n, m) = (n!/m!)*A038231(n-1, m-1) = 4^(n-m)*A008297(n, m) (Lah-triangle). EXAMPLE Triangle begins:       1;       8,     1;      96,    24,    1;    1536,   576,   48,  1;   30720, 15360, 1920, 80, 1;   ... MAPLE # The function BellMatrix is defined in A264428. # Adds (1, 0, 0, 0, ..) as column 0. BellMatrix(n -> 4^n*(n+1)!, 9); # Peter Luschny, Jan 28 2016 MATHEMATICA rows = 8; t = Table[4^n*(n+1)!, {n, 0, rows}]; T[n_, k_] := BellY[n, k, t]; Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *) CROSSREFS Cf. A034177, A038231, A008297. Sequence in context: A114152 A254933 A174503 * A240955 A132056 A051187 Adjacent sequences:  A048783 A048784 A048785 * A048787 A048788 A048789 KEYWORD easy,nonn,tabl AUTHOR EXTENSIONS T(8,4) corrected by Jean-François Alcover, Jun 22 2018 STATUS approved

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Last modified July 28 00:54 EDT 2021. Contains 346316 sequences. (Running on oeis4.)