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 A208678 Row sums of triangle A132623. 5
 1, 3, 8, 29, 157, 1144, 10187, 105600, 1241794, 16287457, 235308853, 3708090433, 63234233743, 1159318599835, 22725352050303, 474059968069223, 10481049913889360, 244727123398669044, 6015958354315188049, 155261610128701766290, 4196413541685139001486 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Triangle T = A132623 is generated by sums of matrix powers of itself such that: T(n,k) = Sum_{j=1..n-k-1} [T^j](n-1,k) with T(n+1,n) = n+1 and T(n,n)=0 for n>=0. Also, column k of triangle T = A132623 obeys the rule: (k+1)*x^(k+1) = Sum_{n>=0} T(n,k) * x^n * (1-x)^(n-k) / Product_{j=k+1..n-1} (1+j*x). LINKS Table of n, a(n) for n=1..21. EXAMPLE Triangle A132623 begins: 0; 1, 0; 1, 2, 0; 3, 2, 3, 0; 14, 8, 3, 4, 0; 87, 46, 15, 4, 5, 0; 669, 338, 102, 24, 5, 6, 0; ... for which this sequence equals the row sums. MATRIX POWER SERIES PROPERTY OF T = A132623: Let T = A132623, then [I - T]^-1 = Sum_{n>=0} T^n yields: 1; 1, 1; 3, 2, 1; 14, 8, 3, 1; 87, 46, 15, 4, 1; 669, 338, 102, 24, 5, 1; ... which equals T shifted up 1 row (but with 1's in the main diagonal). ILLUSTRATE G.F. FOR COLUMN k OF T = A132623: k=0: x = T(1,0)*x*(1-x) + T(2,0)*x^2*(1-x)^2/((1+x)) + T(3,0)*x^3*(1-x)^3/((1+x)*(1+2*x)) + T(4,0)*x^4*(1-x)^4/((1+x)*(1+2*x)*(1+3*x)) +... k=1: 2*x^2 = T(2,1)*x^2*(1-x) + T(3,1)*x^3*(1-x)^2/((1+2*x)) + T(4,1)*x^4*(1-x)^3/((1+2*x)*(1+3*x)) + T(5,1)*x^5*(1-x)^4/((1+2*x)*(1+3*x)*(1+4*x)) +... k=2: 3*x^3 = T(3,2)*x^3*(1-x) + T(4,2)*x^4*(1-x)^2/((1+3*x)) + T(5,2)*x^5*(1-x)^3/((1+3*x)*(1+4*x)) + T(6,2)*x^6*(1-x)^4/((1+3*x)*(1+4*x)*(1+5*x)) +... PROG (PARI) /* Get row sums using the g.f. for columns in triangle A132623: */ {A132623(n, k)=if(n

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Last modified April 22 08:40 EDT 2024. Contains 371893 sequences. (Running on oeis4.)