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A107876
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Triangular matrix T, read by rows, that satisfies: [T^k](n,k) = T(n,k-1) for n>=k>0, or, equivalently, (column k of T^k) = SHIFT_LEFT(column k-1 of T) when zeros above the diagonal are ignored.
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22
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1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 7, 7, 3, 1, 1, 37, 37, 15, 4, 1, 1, 268, 268, 106, 26, 5, 1, 1, 2496, 2496, 975, 230, 40, 6, 1, 1, 28612, 28612, 11100, 2565, 425, 57, 7, 1, 1, 391189, 391189, 151148, 34516, 5570, 707, 77, 8, 1, 1, 6230646, 6230646, 2401365, 544423
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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COMMENTS
| Remarkably, T equals the product of these triangular matrices: T = A107862^-1*A107867 = A107867^-1*A107870 = A107870^-1*A107873. T also satisfies: [T^-1](n,k) = -[T^k](n,k+1) for n>k>=0.
Column m of T^k is the number of subpartitions of the initial terms of the sequence (k-1)+n(m-1)+n(n-1)/2 (ignoring 0's above the diagonal). E.g. column 4 of T^3 is 1,3,15,106,975,.... The sequence above is 2,5,9,14,20,.... subp([]) = 1, subp([2]) = 3, subp([2,5]) = 15, subp([2,5,9]) = 106, etc. The matrix product of T^(k-1) * T computes the number of such subpartitions by looking at the first part index where the subpartition is maxed - for [2,5,9,14,20] the third term (9 maxed) has subp([1,4]) for the first two values (not maxed), times subp([5,11]) for the last two values (possibly maxed). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 26 2006
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FORMULA
| G.f. for column k: 1 = Sum_{j>=0} T(k+j, k)*x^j*(1-x)^(1+(k+j)*(k+j-1)/2-k*(k-1)/2). G.f. for column k of T^m: 1 = Sum_{j>=0} [T^m](k+j, k)*x^j*(1-x)^(m+(k+j)*(k+j-1)/2-k*(k-1)/2) where [T^m] is the m-th matrix power of T, for all m and k>=0. G.f. for column k of T^m: 1 = Sum_{j>=0} [T^m](k+j, k)*x^j/C(x)^(m-j+(k+j)*(k+j-1)/2-k*(k-1)/2) where C(x)=2/(1+sqrt(1-4*x)) is g.f. for A000108 (Catalan numbers).
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EXAMPLE
| G.f. for column 1:
1 = T(1,1)*(1-x)^1 + T(2,1)*x*(1-x)^2 + T(3,1)*x^2*(1-x)^4 + T(4,1)*x^3*(1-x)^7 + T(5,1)*x^4*(1-x)^11 + T(6,1)*x^5*(1-x)^16 +...
= 1*(1-x)^1 + 1*x*(1-x)^2 + 2*x^2*(1-x)^4 + 7*x^3*(1-x)^7 + 37*x^4*(1-x)^11 + 268*x^5*(1-x)^16 +...
G.f. for column 2:
1 = T(2,2)*(1-x)^1 + T(3,2)*x*(1-x)^3 + T(4,2)*x^2*(1-x)^6 + T(5,2)*x^3*(1-x)^10 + T(6,2)*x^4*(1-x)^15 + T(7,2)*x^5*(1-x)^21 +...
= 1*(1-x)^1 + 1*x*(1-x)^3 + 3*x^2*(1-x)^6 + 15*x^3*(1-x)^10 + 106*x^4*(1-x)^15 + 975*x^5*(1-x)^21 +...
Triangle T begins:
1;
1,1;
1,1,1;
2,2,1,1;
7,7,3,1,1;
37,37,15,4,1,1;
268,268,106,26,5,1,1;
2496,2496,975,230,40,6,1,1;
28612,28612,11100,2565,425,57,7,1,1;
391189,391189,151148,34516,5570,707,77,8,1,1; ...
where column 1 of T = SHIFT_LEFT(column 0 of T).
Matrix square T^2 begins:
1;
2,1;
3,2,1;
7,5,2,1;
26,19,7,2,1;
141,104,37,9,2,1;
1034,766,268,61,11,2,1; ...
Compare column 2 of T^2 with column 1 of T.
Matrix inverse begins:
1;
-1,1;
0,-1,1;
0,-1,-1,1;
0,-3,-2,-1,1;
0,-15,-9,-3,-1,1;
0,-106,-61,-18,-4,-1,1; ...
Compare column 1 of T^-1 with column 2 of T and
compare column 2 of T^-1 with column 3 of T^2.
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PROG
| (PARI) {T(n, k)=polcoeff(1-sum(j=0, n-k-1, T(j+k, k)*x^j*(1-x+x*O(x^n))^(1+(k+j)*(k+j-1)/2-k*(k-1)/2)), n-k)} (PARI) {T(n, k, p)=polcoeff(1- sum(j=0, n-k-1, T(j+k, k, p)*x^j*(1-x+x*O(x^n))^(j*(j-1)/2+j*k+p)), n-k)}
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CROSSREFS
| Cf. A107862, A107865, A107867, A107870, A107877 (column 1), A107878 (column 2), A107879 (column 3), A107880 (matrix square), A107884 (matrix cube), A107889 (matrix inverse).
Cf. A115728, A115729.
Sequence in context: A173886 A090441 A155794 * A121554 A011296 A176602
Adjacent sequences: A107873 A107874 A107875 * A107877 A107878 A107879
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KEYWORD
| nonn,tabl,nice
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jun 04 2005
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