A179750
Triangle T(n,k) read by rows. Matrix inverse of A179749.
Original entry on oeis.org
1, -1, 1, 1, -2, 1, -2, 4, -3, 1, 4, -8, 7, -4, 1, -7, 14, -14, 11, -5, 1, 11, -22, 25, -25, 16, -6, 1, -18, 36, -44, 51, -41, 22, -7, 1, 35, -70, 83, -99, 92, -63, 29, -8, 1, -76, 152, -166, 188, -190, 155, -92, 37, -9, 1, 166, -332, 337, -354, 373, -345, 247, -129, 46
Offset: 1
Table begins:
1,
-1,1,
1,-2,1,
-2,4,-3,1,
4,-8,7,-4,1,
-7,14,-14,11,-5,1,
11,-22,25,-25,16,-6,1,
-18,36,-44,51,-41,22,-7,1,
35,-70,83,-99,92,-63,29,-8,1,
-76,152,-166,188,-190,155,-92,37,-9,1,
166,-332,337,-354,373,-345,247,-129,46,-10,1,
-358,716,-693,678,-717,719,-592,376,-175,56,-11,1,
A129273
G.f.: 1-q = Sum_{k>=0} a(k)*q^k * Faq(k+1,q)^2, where Faq(n,q) is the q-factorial of n.
Original entry on oeis.org
1, -1, 2, -7, 26, -95, 344, -1256, 4654, -17470, 66234, -253192, 974992, -3778966, 14729200, -57683066, 226806148, -894791874, 3540105138, -14039128725, 55786507642, -222047783006, 885073034920, -3532110787193, 14110281656038
Offset: 0
Define Faq(n,q) = Product_{i=1..n} (1-q^i)/(1-q) for n>0, Faq(0,q)=1.
Then coefficients of q in a(k)*q^k * Faq(k+1,q)^2 begin as follows:
k=0: 1;
k=1: .. -1, -2,-1;
k=2: ....... 2, 8, 16,.. 20,.. 16,... 8,.... 2;
k=3: ......... -7,-42, -133, -294, -497,. -672, ...;
k=4: ............. 26,. 208,. 884, 2652,. 6266, ...;
k=5: .................. -95, -950,-5035,-18810, ...;
k=6: ........................ 344, 4128, 26144, ...;
k=7: ............................ -1256,-17584, ...;
k=8: .................................... 4654, ...;
Sums cancel along column j for j>1, leaving 1-q.
-
{a(n)=if(n==0,1,polcoeff(1-q- sum(k=0,n-1,a(k)*q^k*prod(j=1,k+1,(1-q^j)/ (1-q+q*O(q^(n-k))))^2),n,q))}
for(n=0,25,print1(a(n),", "))
Showing 1-2 of 2 results.
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