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 A141760 Triangle T, read by rows, where the g.f. of column k in matrix power T^m is given by: 1/(1-x)^m = Sum_{n>=k} [T^m](n,k) * x^(n-k)/(1+x)^{n(n-1)/2 - k(k-1)/2} for k>=0. 4
 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 6, 3, 1, 1, 26, 26, 13, 4, 1, 1, 154, 154, 77, 23, 5, 1, 1, 1188, 1188, 594, 175, 36, 6, 1, 1, 11474, 11474, 5737, 1678, 336, 52, 7, 1, 1, 134432, 134432, 67216, 19579, 3863, 576, 71, 8, 1, 1, 1863168, 1863168, 931584, 270683, 52944, 7731 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 LINKS Table of n, a(n) for n=0..60. FORMULA Matrix powers satisfy: T^m = P(i)^-1 * P(m+i) for all m and i, where P(m) is given by: [P(m)](n,k) = [x^(n-k)] 1/(1-x)^m * (1+x)^{n(n-1)/2 - k(k-1)/2} for n>=k>=0. Let U = unsigned matrix inverse (T^-1) with leftmost column dropped, then U = A107876 where [U^k](n,k) = U(n,k-1) for n>=k>0. G.f. for column k of T: 1/(1-x) = Sum_{n>=0} T(n,k)*x^(n-k)/(1+x)^{n(n-1)/2 - k(k-1)/2}. T(n,k) = 1 - Sum_{j=k..n-1} T(j,k)*(-1)^(n-j)*C(j(j-1)/2 - k(k-1)/2 + n-j-1, n-j) for n>k with T(k,k)=1 for k>=0. G.f. for column k of matrix power T^m: 1/(1-x)^m = Sum_{n>=0} [T^m](n,k)*x^(n-k)/(1+x)^{n*(n-1)/2 - k*(k-1)/2}. [T^m](n,k) = C(m+n-1,n) - Sum_{j=k..n-1} [T^m](j,k)*(-1)^(n-j)*C(j(j-1)/2 - k(k-1)/2 + n-j-1,n-j) for n>k with [T^m](k,k)=1 for k>=0. EXAMPLE Triangle T begins: 1; 1, 1; 1, 1, 1; 2, 2, 1, 1; 6, 6, 3, 1, 1; 26, 26, 13, 4, 1, 1; 154, 154, 77, 23, 5, 1, 1; 1188, 1188, 594, 175, 36, 6, 1, 1; 11474, 11474, 5737, 1678, 336, 52, 7, 1, 1; 134432, 134432, 67216, 19579, 3863, 576, 71, 8, 1, 1; 1863168, 1863168, 931584, 270683, 52944, 7731, 911, 93, 9, 1, 1; ... Matrix square, T^2, begins: 1; 2, 1; 3, 2, 1; 7, 5, 2, 1; 23, 17, 7, 2, 1; 105, 79, 33, 9, 2, 1; 641, 487, 205, 55, 11, 2, 1; 5034, 3846, 1626, 433, 83, 13, 2, 1; ... where g.f. for column k of matrix square T^2 is: 1/(1-x)^2 = Sum_{n>=0} [T^2](n,k)*x^(n-k)/(1+x)^{n(n-1)/2 - k(k-1)/2}. Matrix inverse, T^-1, begins: 1; -1, 1; 0, -1, 1; 0, -1, -1, 1; 0, -2, -2, -1, 1; 0, -7, -7, -3, -1, 1; 0, -37, -37, -15, -4, -1, 1; 0, -268, -268, -106, -26, -5, -1, 1; ... Let U = unsigned T^-1 with leftmost column dropped, then U = A107876 where [U^k](n,k) = U(n,k-1) for n>=k>0. The g.f. for column k of matrix inverse T^-1 is: 1-x = Sum_{n>=0} [T^-1](n,k) * x^(n-k)/(1+x)^{n(n-1)/2 - k(k-1)/2}. MATRIX PRODUCTS: T = P(1)^-1 * P(2) = P(2)^-1 * P(3) = P(m)^-1 * P(m+1); P(1) begins: 1; 1, 1; 2, 2, 1; 8, 7, 3, 1; 57, 42, 16, 4, 1; 638, 386, 130, 29, 5, 1; 9949, 4944, 1471, 299, 46, 6, 1; ... where [P(1)](n,k) = [x^(n-k)] 1/(1-x)*(1+x)^{n(n-1)/2-k(k-1)/2}; P(2) begins: 1; 2, 1; 5, 3, 1; 20, 12, 4, 1; 129, 72, 23, 5, 1; 1268, 630, 187, 38, 6, 1; 17548, 7599, 2063, 392, 57, 7, 1; ... where [P(2)](n,k) = [x^(n-k)] 1/(1-x)^2*(1+x)^{n(n-1)/2-k(k-1)/2}; P(3) begins: 1; 3, 1; 9, 4, 1; 38, 18, 5, 1; 240, 111, 31, 6, 1; 2223, 955, 256, 48, 7, 1; 28672, 11124, 2794, 500, 69, 8, 1; ... where [P(3)](n,k) = [x^(n-k)] 1/(1-x)^3*(1+x)^{n(n-1)/2-k(k-1)/2}. MATHEMATICA T[n_, k_, m_] := T[n, k, m] = If[n

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Last modified May 28 21:57 EDT 2023. Contains 363028 sequences. (Running on oeis4.)