theorem_check / liu_wang_derivative_lag

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

39 sequences

• A100862: Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n…
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … row sums 1, 2, 5, 14, 43, 142, 499, 1850, …
  A100862 (−∞, 0) nonnegative
• A154986: Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.
  1 · 1, 1 · 1, 4, 1 · 1, 11, 11, 1 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A154986 (−∞, 0) nonnegative
• Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.
  1, 7, 1 · 1, 21, 21, 1 · 1, 51, 161, 51, 1 … row sums 9, 44, 265, 1854, 14833, 133496, …
  (−∞, 0) nonnegative
• A144439: Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2, read by rows.
  1 · 1, 1 · 1, 8, 1 · 1, 31, 31, 1 … row sums 1, 2, 10, 64, 532, 5448, 66440, 941056, …
  A144439 (−∞, 0) nonnegative
• A102662: Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).
  1 · 1, 3 · 1, 5, 3 · 1, 7, 11, 3 … row sums 1, 4, 9, 22, 53, 128, 309, 746, …
  A102662 (−∞, 0) nonnegative
• A059473: Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...
  1 · 2, 2 · 4, 10, 4 · 8, 32, 32, 8 … row sums 1, 4, 18, 80, 356, 1584, 7048, 31360, …
  A059473 (−∞, 0) nonnegative
• A143683: Pascal-(1,8,1) array.
  1 · 1, 1 · 1, 10, 1 · 1, 19, 19, 1 … row sums 1, 2, 12, 40, 176, 672, 2752, 10880, …
  A143683 (−∞, 0) nonnegative
• A081582: Pascal-(1,7,1) array.
  1 · 1, 1 · 1, 9, 1 · 1, 17, 17, 1 … row sums 1, 2, 11, 36, 149, 550, 2143, 8136, …
  A081582 (−∞, 0) nonnegative
• A081581: Pascal-(1,6,1) array.
  1 · 1, 1 · 1, 8, 1 · 1, 15, 15, 1 … row sums 1, 2, 10, 32, 124, 440, 1624, 5888, …
  A081581 (−∞, 0) nonnegative
• A395454: Triangle read by rows: T(n, k) = Sum_{j=0..n} C(-j-1, -n-1)*E2(j, n-k), E2 the Eulerian numbers A201637.
  1 · 0, 2 · 0, 6, 2 · 0, 24, 34, 2 … row sums 1, 0, 2, 8, 60, 544, 6040, 79008, …
  A395454 (−∞, 0] nonnegative
• A144438: Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 5, 1 · 1, 14, 14, 1 … row sums 1, 2, 7, 30, 157, 972, 6961, 56660, …
  A144438 (−∞, 0) nonnegative
• A144445: Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 23, 1 · 1, 206, 206, 1 … row sums 1, 2, 25, 414, 9697, 293808, 10938775, …
  A144445 (−∞, 0) nonnegative
• A334218: Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.
  1 · 1 · 0, 2 · 0, 3, 3 … row sums 1, 1, 2, 6, 24, 120, 720, 5040, …
  A334218 (−∞, 0] nonnegative
• A249248: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 3 · 1, 7, 3 · 1, 12, 22, 3 … row sums 1, 4, 11, 38, 131, 490, 1897, 7714, …
  A249248 (−∞, 0) nonnegative
• A081580: Pascal-(1,5,1) array.
  1 · 1, 1 · 1, 7, 1 · 1, 13, 13, 1 … row sums 1, 2, 9, 28, 101, 342, 1189, 4088, …
  A081580 (−∞, 0) nonnegative
• A144441: Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 14, 1 · 1, 83, 83, 1 … row sums 1, 2, 16, 168, 2416, 44160, 981184, …
  A144441 (−∞, 0) nonnegative
• A154233: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.
  1 · 1, 1 · 1, 7, 1 · 1, 19, 19, 1 … row sums 1, 2, 9, 40, 251, 1662, 13615, 118640, …
  A154233 (−∞, 0) nonnegative
• A143685: Pascal-(1,9,1) array.
  1 · 1, 1 · 1, 11, 1 · 1, 21, 21, 1 … row sums 1, 2, 13, 44, 205, 806, 3457, 14168, …
  A143685 (−∞, 0) nonnegative
• A008288: Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
  1 · 1, 1 · 1, 3, 1 · 1, 5, 5, 1 … row sums 1, 2, 5, 12, 29, 70, 169, 408, …
  A008288 [−3−2√2, −3+2√2] nonnegative
• A154227: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.
  1 · 1, 1 · 1, 8, 1 · 1, 19, 19, 1 … row sums 1, 2, 10, 40, 230, 1300, 9040, 64880, …
  A154227 (−∞, 0) nonnegative
• A144436: Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 1, and j = 4, read by rows.
  1 · 1, 1 · 1, 8, 1 · 1, 23, 23, 1 … row sums 1, 2, 10, 48, 280, 1872, 14224, 121280, …
  A144436 (−∞, 0) nonnegative
• A144443: Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 20, 1 · 1, 159, 159, 1 … row sums 1, 2, 22, 320, 6532, 171752, 5535256, …
  A144443 (−∞, 0) nonnegative
• A266213: Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*bi…
  1 · 1 · 1, 2 · 1, 4, 2 … row sums 1, 1, 3, 7, 17, 41, 99, 239, …
  A266213 (−∞, 0) nonnegative
• A154228: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1), read by rows.
  1 · 1, 1 · 1, 16, 1 · 1, 47, 47, 1 … row sums 1, 2, 18, 96, 1182, 11100, 187680, …
  A154228 (−∞, 0) nonnegative
• A144442: Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 17, 1 · 1, 118, 118, 1 … row sums 1, 2, 19, 238, 4141, 92292, 2512589, …
  A144442 (−∞, 0) nonnegative
• A122542: Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
  1 · 0, 1 · 0, 2, 1 · 0, 2, 4, 1 … row sums 1, 1, 3, 7, 17, 41, 99, 239, …
  A122542 (−∞, 0] nonnegative
• A271697: Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j-1,-n-1)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.
  1 · 0, 1 · 0, 1, 1 · 0, 1, 7, 1 … row sums 1, 0, 1, 2, 9, 44, 265, 1854, …
  A271697 (−∞, 0] nonnegative
• A121574: Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
  1 · 2, 1 · 4, 5, 1 · 8, 16, 8, 1 … row sums 1, 3, 10, 33, 109, 360, 1189, 3927, …
  A121574 (−∞, 0) nonnegative
• A116412: Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
  1 · 3, 1 · 6, 6, 1 · 12, 21, 9, 1 … row sums 1, 4, 13, 43, 142, 469, 1549, 5116, …
  A116412 (−∞, 0) nonnegative
• A081578: Pascal-(1,3,1) array.
  1 · 1, 1 · 1, 5, 1 · 1, 9, 9, 1 … row sums 1, 2, 7, 20, 61, 182, 547, 1640, …
  A081578 (−∞, 0) nonnegative
• delannoy
  1 · 1, 1 · 1, 3, 1 · 1, 5, 5, 1 … row sums 1, 2, 5, 12, 29, 70, 169, 408, …
  [−3−2√2, −3+2√2] nonnegative
• A035607: Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).
  1 · 1, 2 · 1, 4, 2 · 1, 6, 8, 2 … row sums 1, 3, 7, 17, 41, 99, 239, 577, …
  A035607 (−∞, 0) nonnegative
• A249252: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 4 · 1, 8, 8 · 1, 12, 32, 16 … row sums 1, 5, 17, 61, 217, 773, 2753, 9805, …
  A249252 (−∞, 0) nonnegative
• A249251: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 3 · 1, 6, 6 · 1, 9, 21, 12 … row sums 1, 4, 13, 43, 142, 469, 1549, 5116, …
  A249251 (−∞, 0) nonnegative
• A249250: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 3 · 1, 6, 3 · 1, 9, 15, 3 … row sums 1, 4, 10, 28, 76, 208, 568, 1552, …
  A249250 (−∞, 0) nonnegative
• A113413: A Riordan array of coordination sequences.
  1 · 2, 1 · 2, 4, 1 · 2, 8, 6, 1 … row sums 1, 3, 7, 17, 41, 99, 239, 577, …
  A113413 (−∞, 0) nonnegative
• A081579: Pascal-(1,4,1) array.
  1 · 1, 1 · 1, 6, 1 · 1, 11, 11, 1 … row sums 1, 2, 8, 24, 80, 256, 832, 2688, …
  A081579 (−∞, 0) nonnegative
• A144440: Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 11, 1 · 1, 54, 54, 1 … row sums 1, 2, 13, 110, 1249, 17816, 306619, …
  A144440 (−∞, 0) nonnegative
• A081577: Pascal-(1,2,1) array read by antidiagonals.
  1 · 1, 1 · 1, 4, 1 · 1, 7, 7, 1 … row sums 1, 2, 6, 16, 44, 120, 328, 896, …
  A081577 (−∞, 0) nonnegative