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A308737 Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n,k)_1 where (n,k)_1 is the 1-tiered binomial coefficient. 0
1, 1, 3, 1, 8, 7, 1, 17, 31, 15, 1, 34, 96, 94, 31, 1, 67, 258, 382, 253, 63, 1, 132, 645, 1280, 1275, 636, 127, 1, 261, 1545, 3845, 5115, 3831, 1531, 255, 1, 518, 3598, 10766, 17920, 17906, 10738, 3578, 511, 1, 1031, 8212, 28700, 57358, 71666, 57316, 28652, 8185, 1023 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..54.

Michael E. Hoffman, (Poly)logarithmic Integrals and Multiple Zeta Values, Number Theory Talk, Max-Planck-Institut für Mathematik, Bonn, 20 June 2018. See Slide 29.

Markus Kuba, A note on logarithmic integrals and tiered binomial coefficients, arXiv:1906.08347 [math.CO], 2019. See recurrence p. 8 and 10.

FORMULA

(n,m)_1 = ((n-1,m)_1 + (n,m-1)_1 + (n,m-1)_0)/2, (0,0)_1 = 1/2, where (n,m)_0 is binomial(n,m) (recurrence for the square array presentation of the regular triangle).

Scaled coefficients satisfy T(n,0) = 1, T(n,k) = T(n-1,k) + T(n,k-1) + 2^(n+1)*C(n-1,k-1). - Charlie Neder, Jun 21 2019

G.f.: (1-x)/((1-x-y)(2-x-y)). - Jean-François Alcover, Jun 21 2019

EXAMPLE

Triangle of 1-tiered binomial coefficients:

  1/2,

  1/4,    3/4,

  1/8,     1,    7/8,

  1/16,  17/16, 31/16, 15/16,

  1/32,  17/16,   3,   47/16, 31/32,

  ...

Scaled triangle after multiplying each row by 2^(n+1):

  1,

  1,  3,

  1,  8,  7,

  1, 17, 31, 15,

  1, 34, 96, 94, 31,

  ...

MATHEMATICA

rows = 10;

cc = CoefficientList[# + O[y]^rows, y]& /@ CoefficientList[(1-x)/((1-x-y)* (2-x-y)) + O[x]^rows, x];

T[n_, m_, 1] := cc[[n-m+1, m+1]];

Table[2^(n+1) Table[T[n, m, 1], {m, 0, n}], {n, 0, rows-1}] (* Jean-François Alcover, Jun 21 2019 *)

PROG

(PARI) T(n, m) = if ((n==0) && (m==0), 1/2, binomial(n+m-1, m-1) - (binomial(n+m, n)/2 - binomial(n+m-1, n-1))/2^(n+m));

TT(n, k) = T(n-k, k);

tabls(nn) = for (n=0, nn, for (k=0, n, print1(2^(n+1)*TT(n, k), ", ")));

CROSSREFS

Cf. A007318 (Pascal triangle: 0-tiered binomial coefficient).

Sequence in context: A286416 A005295 A077897 * A257142 A270861 A208656

Adjacent sequences:  A308734 A308735 A308736 * A308738 A308739 A308740

KEYWORD

nonn,tabl

AUTHOR

Michel Marcus, Jun 21 2019

STATUS

approved

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Last modified February 21 20:35 EST 2020. Contains 332111 sequences. (Running on oeis4.)