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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 Table of n, a(n) for n=1..36. 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: A347111 A254933 A174503 * A240955 A132056 A051187 Adjacent sequences: A048783 A048784 A048785 * A048787 A048788 A048789 KEYWORD easy,nonn,tabl AUTHOR Wolfdieter Lang EXTENSIONS T(8,4) corrected by Jean-François Alcover, Jun 22 2018 STATUS approved

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Last modified August 9 11:10 EDT 2024. Contains 375040 sequences. (Running on oeis4.)