theorem_check / ma_wang_interval

Theorem check of the node's own rule. All operations

20 sequences

• A099093: Riordan array (1, 3+3x).
  1 · 0, 3 · 0, 3, 9 · 0, 0, 18, 27 … row sums 1, 3, 12, 45, 171, 648, 2457, 9315, …
  A099093 (−4/3, 0] nonnegative
• A225472: Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  1 · 2, 3 · 4, 21, 18 · 8, 117, 270, 162 … row sums 1, 5, 43, 557, 9643, 208685, 5419243, …
  A225472 (−1, 0) nonnegative
• A030528: Triangle read by rows: T(n,k) = binomial(k,n-k).
  1 · 1, 1 · 0, 2, 1 · 0, 1, 3, 1 … row sums 1, 2, 3, 5, 8, 13, 21, 34, …
  A030528 (−4, 0] nonnegative
• A163626: Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).
  1 · 1, -1 · 1, -3, 2 · 1, -7, 12, -6 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  A163626 (0, 1]
• A131689: Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.
  1 · 0, 1 · 0, 1, 2 · 0, 1, 6, 6 … row sums 1, 1, 3, 13, 75, 541, 4683, 47293, …
  A131689 (−1, 0] nonnegative
• A026729: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
  1 · 0, 1 · 0, 1, 1 · 0, 0, 2, 1 … row sums 1, 1, 2, 3, 5, 8, 13, 21, …
  A026729 (−4, 0] nonnegative
• A134991: Triangle of Ward numbers T(n,k) read by rows.
  1 · 1, 3 · 1, 10, 15 · 1, 25, 105, 105 … row sums 1, 4, 26, 236, 2752, 39208, 660032, …
  A134991 (−1, 0) nonnegative
• A convolution triangle of numbers obtained from A036224.
  336, 42, 1 · 4536, 1113, 63, 1 … row sums 379, 5713, 80032, 1080325, 14354557, …
  [−36, 0) nonnegative
• A112493: Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.
  1 · 1, 1 · 1, 4, 3 · 1, 11, 25, 15 … row sums 1, 2, 8, 52, 472, 5504, 78416, 1320064, …
  A112493 [−1, 0) nonnegative
• A099097: Riordan array (1, 3+x).
  1 · 0, 3 · 0, 1, 9 · 0, 0, 6, 27 … row sums 1, 3, 10, 33, 109, 360, 1189, 3927, …
  A099097 (−4/9, 0] nonnegative
• A028246: Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.
  1 · 1, 1 · 1, 3, 2 · 1, 7, 12, 6 … row sums 1, 2, 6, 26, 150, 1082, 9366, 94586, …
  A028246 [−1, 0) nonnegative
• A225476: Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  1 · 1, 1 · 1, 4, 2 · 1, 13, 18, 6 … row sums 1, 2, 7, 38, 277, 2522, 27547, 351038, …
  A225476 (−2, 0) nonnegative
• A112486: Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.
  1 · 1, 1 · 2, 5, 3 · 6, 26, 35, 15 … row sums 1, 2, 10, 82, 938, 13778, 247210, …
  A112486 [−1, 0) nonnegative
• A113953: A Jacobsthal triangle.
  1 · 0, 1 · 0, 2, 1 · 0, 0, 4, 1 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  A113953 (−8, 0] nonnegative
• A113214: Riordan array (1+2x,x(1+x)).
  1 · 2, 1 · 0, 3, 1 · 0, 2, 4, 1 … row sums 1, 3, 4, 7, 11, 18, 29, 47, …
  A113214 (−4, 0] nonnegative
• A063967: Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
  1 · 1, 1 · 2, 3, 1 · 3, 7, 5, 1 … row sums 1, 2, 6, 16, 44, 120, 328, 896, …
  A063967 (−5, −1] nonnegative
• A019538: Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
  1 · 1, 2 · 1, 6, 6 · 1, 14, 36, 24 … row sums 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  A019538 (−1, 0) nonnegative
• A269939: Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.
  1 · 0, 1 · 0, 1, 3 · 0, 1, 10, 15 … row sums 1, 1, 4, 26, 236, 2752, 39208, 660032, …
  A269939 (−1, 0] nonnegative
• A225473: Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  1 · 3, 4 · 9, 40, 32 · 27, 316, 672, 384 … row sums 1, 7, 81, 1399, 32289, 931687, …
  A225473 (−1, 0) nonnegative
• A099040: Riordan array (1, 2+2x).
  1 · 0, 2 · 0, 2, 4 · 0, 0, 8, 8 … row sums 1, 2, 6, 16, 44, 120, 328, 896, …
  A099040 (−2, 0] nonnegative