This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A107876 #42 Feb 11 2023 13:53:34
%S A107876 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,
%T A107876 2496,2496,975,230,40,6,1,1,28612,28612,11100,2565,425,57,7,1,1,
%U A107876 391189,391189,151148,34516,5570,707,77,8,1,1,6230646,6230646,2401365,544423
%N A107876 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.
%C A107876 Remarkably, T equals the product of these triangular matrices: T = A107862^-1*A107867 = A107867^-1*A107870 = A107870^-1*A107873; reversing the order of these products yields triangle A101479.
%C A107876 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). - _Franklin T. Adams-Watters_, Jun 26 2006
%C A107876 T(n,k) is the number of Dyck paths whose sequence of ascent lengths is exactly k,k+1,...,n, for example the T(4,3) = 3 paths are UUUdUUUUd^6, UUUddUUUUd^5 and UUUdddUUUUd^4. - _David Scambler_, May 30 2012
%H A107876 Alois P. Heinz, <a href="/A107876/b107876.txt">Rows n = 0..50, flattened</a>
%F A107876 G.f. for column k of T^m, the m-th matrix power of this triangle T:
%F A107876 (1) 1 = Sum_{j>=0} T(k+j, k) * x^j * (1-x)^(1+(k+j)*(k+j-1)/2-k*(k-1)/2) for m=1.
%F A107876 (2) 1 = Sum_{j>=0} [T^m](k+j, k)*x^j*(1-x)^(m+(k+j)*(k+j-1)/2-k*(k-1)/2) for all m and k>=0.
%F A107876 (3) 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).
%F A107876 Matrix inverse of this triangle T satisfies:
%F A107876 (4) [T^-1](n,k) = -[T^k](n,k+1) for n>k>=0.
%e A107876 G.f. for column 1:
%e A107876 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 +...
%e A107876 = 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 +...
%e A107876 G.f. for column 2:
%e A107876 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 +...
%e A107876 = 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 +...
%e A107876 Triangle T begins:
%e A107876 1;
%e A107876 1, 1;
%e A107876 1, 1, 1;
%e A107876 2, 2, 1, 1;
%e A107876 7, 7, 3, 1, 1;
%e A107876 37, 37, 15, 4, 1, 1;
%e A107876 268, 268, 106, 26, 5, 1, 1;
%e A107876 2496, 2496, 975, 230, 40, 6, 1, 1;
%e A107876 28612, 28612, 11100, 2565, 425, 57, 7, 1, 1;
%e A107876 391189, 391189, 151148, 34516, 5570, 707, 77, 8, 1, 1; ...
%e A107876 where column 1 of T = SHIFT_LEFT(column 0 of T).
%e A107876 Matrix square T^2 begins:
%e A107876 1;
%e A107876 2, 1;
%e A107876 3, 2, 1;
%e A107876 7, 5, 2, 1;
%e A107876 26, 19, 7, 2, 1;
%e A107876 141, 104, 37, 9, 2, 1;
%e A107876 1034, 766, 268, 61, 11, 2, 1; ...
%e A107876 Compare column 2 of T^2 with column 1 of T.
%e A107876 Matrix inverse begins:
%e A107876 1;
%e A107876 -1, 1;
%e A107876 0, -1, 1;
%e A107876 0, -1, -1, 1;
%e A107876 0, -3, -2, -1, 1;
%e A107876 0, -15, -9, -3, -1, 1;
%e A107876 0, -106, -61, -18, -4, -1, 1; ...
%e A107876 Compare column 1 of T^-1 with column 2 of T and
%e A107876 compare column 2 of T^-1 with column 3 of T^2.
%t A107876 max = 10;
%t A107876 A107862 = Table[Binomial[If[n<k, 0, n*(n-1)/2 - k*(k - 1)/2 + n-k], n-k], {n, 0, max}, {k, 0, max}];
%t A107876 A107867 = Table[Binomial[If[n<k, 0, n*(n-1)/2 - k*(k - 1)/2 + n-k+1], n-k], {n, 0, max}, {k, 0, max}];
%t A107876 t = Inverse[A107862].A107867;
%t A107876 Table[t[[n, k]], {n, 1, max+1}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 12 2012, after first comment, fixed by _Vaclav Kotesovec_, Jun 13 2018 *)
%o A107876 (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)}
%o A107876 for(n=0,10,for(k=0,n,print1(T(n,k),", ")); print(""))
%o A107876 (PARI) /* Print the Triangular Matrix to the Power p: */
%o A107876 {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)}
%o A107876 for(n=0,10,for(k=0,n,print1(T(n,k,1),", ")); print(""))
%Y A107876 Cf. A107862, A107865, A107867, A107870, A107877 (column 1), A107878 (column 2), A107879 (column 3), A107880 (matrix square), A107884 (matrix cube), A107889 (matrix inverse).
%Y A107876 Cf. A115728, A115729, A101479 (dual triangle).
%Y A107876 T(2n,n) gives A300954.
%K A107876 nonn,tabl,nice
%O A107876 0,7
%A A107876 _Paul D. Hanna_, Jun 04 2005