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 A230448 #4 Oct 19 2013 10:45:55 %S A230448 1,1,0,1,1,3,1,2,4,5,1,3,6,9,16,1,4,9,15,25,39,1,5,13,24,40,64,105,1, %T A230448 6,18,37,64,104,169,272,1,7,24,55,101,168,273,441,715,1,8,31,79,156, %U A230448 269,441,714,1156,1869,1,9,39,110,235,425,710,1155,1870,3025,4896 %N A230448 T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = A226205(n+1), n >= 0 and 0 <= k <= n. %C A230448 Triangle T(n, k) is related to the Kn1p sums of the ‘Races with Ties’ triangle A035317. See A230447 for the Kn1p sums and A180662 for the definitions of these sums. %C A230448 The row sums equal ((-1)^n*3*A083581(n) + A022379(2*n+2))/15. %C A230448 Note that the partial fraction expansion of the G.f. of the terms in the n-th row of the square array Tsq(n, k) = T(n+k, k) is related to A014334, the exponential convolution of the Fibonacci numbers with themselves, and to A000032, the Lucas numbers. %F A230448 T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = F(n+2) * F(n-1) = A226205(n+1) with F(n) = A000045(n), the Fibonacci numbers, n >= 0 and 0 <= k <= n. %F A230448 T(n, k) = sum(A035317(n+k-p-2, p), p=0..k), n >= 0 and 0 <= k <= n. %F A230448 T(n+p+2, p-2) = A080239(n+2*p-1) - sum(A035317(n-k+p-1, k+p-1), k=0..floor(n/2)), n >= 0 and p >= 2. %F A230448 The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0. %F A230448 Tsq(n, k) = sum(Tsq(n-1, i), i=0..k), n >= 1 and k >= 0, with Tsq(0, k) = A226205(k+1). %F A230448 The two G.f.’s given below generate the terms in the n-th row of the square array Tsq(n, k). The remarkable second G.f. is the partial fraction expansion of the first G.f.. %F A230448 G.f.: 1/((1-x)^(n-2)*(1+x)*(x^2-3*x+1)), n >= 0. %F A230448 G.f.: sum((-1)^(n+k-1)*A014334(k+2)/(2^(k+2)*(x-1)^(n-k-2)), k=0..n-3) + 1/(5*2^(n-2)*(1+x)) + (A000032(n+1) - A000032(n-1)*x)/(5*(x^2-3*x+1)), n >= 0. %e A230448 The first few rows of triangle T(n, k), n >= 0 and 0 <= k <= n. %e A230448 n/k 0 1 2 3 4 5 6 7 %e A230448 ------------------------------------------------ %e A230448 0| 1 %e A230448 1| 1, 0 %e A230448 2| 1, 1, 3 %e A230448 3| 1, 2, 4, 5 %e A230448 4| 1, 3, 6, 9, 16 %e A230448 5| 1, 4, 9, 15, 25, 39 %e A230448 6| 1, 5, 13, 24, 40, 64, 105 %e A230448 7| 1, 6, 18, 37, 64, 104, 169, 272 %e A230448 The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0. %e A230448 n/k 0 1 2 3 4 5 6 7 %e A230448 ------------------------------------------------ %e A230448 0| 1, 0, 3, 5, 16, 39, 105, 272 %e A230448 1| 1, 1, 4, 9, 25, 64, 169, 441 %e A230448 2| 1, 2, 6, 15, 40, 104, 273, 714 %e A230448 3| 1, 3, 9, 24, 64, 168, 441, 1155 %e A230448 4| 1, 4, 13, 37, 101, 269, 710, 1865 %e A230448 5| 1, 5, 18, 55, 156, 425, 1135, 3000 %e A230448 6| 1, 6, 24, 79, 235, 660, 1795, 4795 %e A230448 7| 1, 7, 31, 110, 345, 1005, 2800, 7595 %p A230448 T := proc(n, k) option remember: if k=0 then return(1) elif k=n then return(combinat[fibonacci](n+2)*combinat[fibonacci](n-1)) else procname(n-1, k-1) + procname(n-1, k) fi: end: seq(seq(T(n, k), k=0..n), n=0..10); # End first program. %p A230448 T := proc(n, k): add(A035317(n+k-p-2, p), p=0..k) end: A035317 := proc(n, k): add((-1)^(i+k) * binomial(i+n-k+1, i), i=0..k) end: seq(seq(T(n, k), k=0..n), n=0..10); # End second program. %Y A230448 Cf. (Triangle columns) A000012, A001477, A152950, A226205, A007598, A001654, A064831, A080145 %K A230448 nonn,easy,tabl %O A230448 0,6 %A A230448 _Johannes W. Meijer_, Oct 19 2013