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 A271238 Triangle of numbers where T(n,k) is the number of k-dimensional faces on a completely truncated n-dimensional simplex, 0 <= k <= n. 0
 1, 2, 1, 3, 3, 1, 6, 12, 8, 1, 10, 30, 30, 10, 1, 15, 60, 80, 45, 12, 1, 21, 105, 175, 140, 63, 14, 1, 28, 168, 336, 350, 224, 84, 16, 1, 36, 252, 588, 756, 630, 336, 108, 18, 1, 45, 360, 960, 1470, 1512, 1050, 480, 135, 20, 1, 55, 495, 1485, 2640, 3234, 2772, 1650, 660, 165, 22, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The complete truncation of a 1-dimensional segment is also a 1-dimensional segement (rather than degenerating to a point). LINKS FORMULA G.f. for rows (n > 0): (((x+1)^n-1)*(x+n+2))/x-n-binomial(n+1,2)*(x+1). O.g.f.: (1/(1-(x+1)*y)^2-(x+1)/(1-y)^2)/x + 1/((1-(x+1)*y)*(1-y))+1+y*(x+1)*(1-1/(1-y)^3). E.g.f.: ((x+1)*(z+1)+1)*exp(z)*(exp(x*z)-1)/x + 1 - (x+1)*z*((z+2)*exp(z)-2)/2. EXAMPLE Triangle begins: 1; 2, 1; 3, 3, 1; 6, 12, 8, 1; 10, 30, 30, 10, 1; ... Row 2 describes a triangle. Row 3 describes an octahedron. MATHEMATICA Flatten[Table[   CoefficientList[    D[((x + 1) (z + 1) + 1) Exp[z] (Exp[x z] - 1)/x +       1 - (x + 1) z ((z + 2)*Exp[z] - 2)/2, {z, k}] /. z -> 0, x], {k, 0,    10}]] CROSSREFS Cf. A259477 (partially-truncated simplex). Sequence in context: A165007 A284979 A127123 * A186740 A103525 A294432 Adjacent sequences:  A271235 A271236 A271237 * A271239 A271240 A271241 KEYWORD nonn,tabl AUTHOR Vincent J. Matsko, Apr 02 2016 STATUS approved

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Last modified January 19 23:47 EST 2022. Contains 350467 sequences. (Running on oeis4.)