%I A087127
%S A087127 1,1,2,1,1,8,19,18,6,1,26,163,432,564,360,90,1,80,1135,6354,18078,28800,
%T A087127 26100,12600,2520,1,242,7291,77400,405060,1210680,2211570,2520000,
%U A087127 1751400,680400,113400,1,728,45199,862218,7667646,38350080,118848420
%N A087127 This table shows the sobalian coefficients of combinatorial formulae 
               needed for generating the sequential sums of p-th powers of triangular 
               numbers. The p-th row (p>=1) contains a(i,p) for i=1 to 2*p-1, where 
               a(i,p) satisfies Sum_{i=1..n} C(i+1,2)^p = 3 * C(n+2,3) * Sum_{i=1..2*p-1} 
               a(i,p) * C(n-1,i-1)/(i+2).
%H A087127 A. F. Labossiere, <a href="http://membres.lycos.fr/labos2/resume.html">
               Sobalian Coefficients</a>.
%H A087127 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/
               ">Miscellaneous</a>.
%H A087127 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">
               Les coefficients sobaliens</a>.
%F A087127 a(1, p) = 1, a(2, p) = 3^(p-1)-1, a(3, p) = 3^(p-1)*[2^(p-1)-2]+1, ..., 
               a(2*p-3, p) = [ (6*p^4-20*p^3+21*p^2-7*p)*(2*p-4)! ]/[3*2^(p-1)], 
               a(2*p-2, p) = [ (p^2-p)*(2*p-3)! ]/2^(p-2), a(2*p-1, p) = [ (p-1)*(2*p-3)! 
               ]/2^(p-2).
%F A087127 a(i, p) = Sum_{k=1..[2*i+1+(-1)^(i-1)]/4} [ C(i-1, 2*k-2)*C(i-2*k+3, 
               i-2*k+1)^(p-1) -C(i-1, 2*k-1)*C(i-2*k+2, i-2*k)^(p-1) ]
%e A087127 Row 3 contains 1,8,19,18,6, so Sum_{i=1..n} C(i+1,2)^3 = (n+2) * C(n+1,
               2) * [ a(1,3)/3 + a(2,3)*C(n-1,1)/4 + a(3,3)*C(n-1,2)/5 + a(4,3)*C(n-1,
               3)/6 + a(5,3)*C(n-1,4)/7 ] = [ (n+2)*(n+1)*n/2 ] * [ 1/3 + (8/4)*C(n-1,
               1) + (19/5)*C(n-1,2) + (18/6)*C(n-1,3) + (6/7)*C(n-1,4). Cf. A085438 
               for more details.
%Y A087127 Cf. A000292, A024166, A024166, A085438, A085439, A085440, A085441, A085442, 
               A087107, A000332, A086020, A086021, A086022, A087108, A000389, A086023, 
               A086024, A087109, A000579, A086025, A086026, A087110, A000580, A086027, 
               A086028, A087111, A027555, A086029, A086030.
%Y A087127 Sequence in context: A156901 A167400 A165889 this_sequence A144946 A157109 
               A167015
%Y A087127 Adjacent sequences: A087124 A087125 A087126 this_sequence A087128 A087129 
               A087130
%K A087127 easy,nonn,tabf
%O A087127 1,3
%A A087127 Andre F. Labossiere (boronali(AT)laposte.net), Aug 11 2003
%E A087127 Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Aug 16 2003