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 A189391 The minimum possible value for the apex of a triangle of numbers whose base consists of a permutation of the numbers 0 to n, and each number in a higher row is the sum of the two numbers directly below it. 14
 0, 1, 3, 8, 19, 44, 98, 216, 467, 1004, 2134, 4520, 9502, 19928, 41572, 86576, 179587, 372044, 768398, 1585416, 3263210, 6711176, 13775068, 28255568, 57863214, 118430584, 242061468, 494523536, 1009105372, 2058327344, 4194213448 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS This is the Riordan transform of A000217 (triangular numbers) with the Riordan matrix (of the Bell type) A053121 (inverse of the Chebyshev S Bell matrix). See the resulting formulae below. - Wolfdieter Lang, Feb 18 2017. Conjecture: a(n) is also half the sum of the "cuts-resistance" (see A319416, A319420, A319421) of all binary vectors of length n (see Lenormand, page 4). - N. J. A. Sloane, Sep 20 2018 LINKS Nathaniel Johnston, Table of n, a(n) for n = 0..1000 F. Disanto and S. Rinaldi, Symmetric convex permutominoes and involutions, PU. M. A., Vol. 22 (2011), No. 1, pp. 39-60. - From N. J. A. Sloane, May 04 2012 Steven R. Finch, How far might we walk at random?, arXiv:1802.04615 [math.HO], 2018. Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. - N. J. A. Sloane, Sep 20 2018 FORMULA If n even, a(n) = (n+1/2)*binomial(n,n/2) - 2^(n-1); if n odd, a(n) = ((n+1)/2)*binomial(n+1,(n+1)/2) - 2^(n-1). - N. J. A. Sloane, Nov 01 2018 a(n) = Sum_{k=0..floor((n-1)/2)} (2*n-4*k-1)*binomial(n,k). G.f.: (2*x+sqrt(1-4*x^2)-1) / (2*(2*x-1)^2). - Alois P. Heinz, Feb 09 2012 a(n) ~ 2^n * (sqrt(2n/Pi)- 1/2). - Vaclav Kotesovec, Mar 16 2014 (formula simplified by Lewis Chen, May 25 2017) D-finite with recurrence n*a(n) + (n-5)*a(n-1) + 2*(-5*n+6)*a(n-2) + 4*(-n+8)*a(n-3) + 24*(n-3)*a(n-4) = 0. - R. J. Mathar, Jan 04 2017 From Wolfdieter Lang, Feb 18 2017:(Start) a(n) = Sum_{m=0..n} A053121(n, m)*A000217(m), n >= 0. G.f.: c(x^2)*Tri(x*c(x^2)), with c and Tri the g.f. of A000108 and A000217, respectively. See the explicit form of the g.f. given above by Alois P. Heinz. (End) 2*a(n) = A152548(n)-2^n. - R. J. Mathar, Jun 17 2021 EXAMPLE For n = 4 consider the triangle: ....19 ...8 11 ..5 3 8 .4 1 2 6 3 1 0 2 4 This triangle has 19 at its apex and no other such triangle with the numbers 0 - 4 on its base has a smaller apex value, so a(4) = 19. MAPLE a:=proc(n)return add((2*n-4*k-1)*binomial(n, k), k=0..floor((n-1)/2)): end: seq(a(n), n=0..50); MATHEMATICA CoefficientList[Series[(2*x+Sqrt[1-4*x^2]-1) / (2*(2*x-1)^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 16 2014 *) PROG (PARI) A189391(n)=sum(i=0, (n-1)\2, (2*n-4*i-1)*binomial(n, i)) \\ M. F. Hasler, Jan 24 2012 (Magma) m:=30; R:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!((2*x+Sqrt(1-4*x^2)-1)/(2*(2*x-1)^2))); // G. C. Greubel, Aug 24 2018 CROSSREFS Cf. A066411, A099325, A189390, A189162, A000217, A053121, A281862. Cf. also A319416, A319420, A319421. Sequence in context: A332719 A326599 A121551 * A281812 A077850 A097550 Adjacent sequences: A189388 A189389 A189390 * A189392 A189393 A189394 KEYWORD easy,nonn AUTHOR Nathaniel Johnston, Apr 20 2011 STATUS approved

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Last modified April 20 02:14 EDT 2024. Contains 371798 sequences. (Running on oeis4.)