A211246 - OEIS

%I #22 Oct 08 2018 09:01:59

%S 4,3,6,8,5,5,8,19,12,7,14,7,10,34,44,16,9,27,30,9,12,53,104,85,20,11,

%T 44,77,55,11,14,76,200,259,146,24,13,65,156,182,91,13,16,103,340,606,

%U 560,231,28,15,90,275,450,378,140,15,18,134,532,1210,1572,1092,344,32

%N Irregular triangle read by rows: numbers a_m(i) related to Bernoulli and Euler polynomials.

%C See the Sun (2008) reference for the (complicated) precise definition.

%H Zhi-Wei Sun, <a href="https://arxiv.org/abs/math/0404385">On sums of binomial coefficients and their applications</a>, arXiv:math/0404385 [math.NT], 2004-2008.

%H Zhi-Wei Sun, <a href="https://doi.org/10.1016/j.disc.2007.08.046">On sums of binomial coefficients and their applications</a>, Discrete Math. 308 (2008), no. 18, 4231--4245. MR2427754(2010d:05002).

%e Triangle begins:

%e 4

%e 3

%e 6 8

%e 5 5

%e 8 19 12

%e 7 14 7

%e 10 34 44 16

%e 9 27 30 9

%e 12 53 104 85 20

%e 11 44 77 55 11

%e 14 76 200 259 146 24

%e ...

%t c[m_, i_] := Binomial[m-1-i, i] + 4 Binomial[m-i, i-1];

%t d[m_, i_] := Binomial[m-i, i] m/(m -i);

%t a[m_, i_] := If[EvenQ[m], c[m, i], d[m, i]];

%t Table[a[m, i], {m, 1, 16}, {i, 1, m/2}] // Flatten (* _Jean-François Alcover_, Oct 08 2018, from PARI *)

%o (PARI)

%o c(m,i) = binomial(m-1-i,i) + 4*binomial(m-i,i-1);

%o d(m,i) = binomial(m-i,i)*m / (m-i);

%o a(m,i) = if ( m%2 == 0, c(m,i), d(m,i) );

%o for (m=1,16, for (i=1,floor(m/2), print1( a(m,i), ", "))) \\ _Hugo Pfoertner_, Nov 01 2017

%Y Cf. A034807, A211247, A267633.

%K nonn,tabf

%O 2,1

%A _N. J. A. Sloane_, Apr 05 2012

%E More terms from _Hugo Pfoertner_, Nov 01 2017