theorem_check

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

304 sequences

• A225117: Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.
  1 · 2, 1 · 4, 13, 1 · 8, 93, 60, 1 … row sums 1, 3, 18, 162, 1944, 29160, 524880, …
  A225117 (−∞, 0) nonnegative
• A123202: Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.
  1 · 1, -2 · 2, -8, 7 · 6, -36, 63, -34 … row sums 1, -1, 1, -1, 1, -1, 1, -1, …
  A123202 (0, 1)
• A039991: Triangle of coefficients of cos(x)^n in polynomial for cos(nx).
  1 · 1 · 2, 0, -1 · 4, 0, -3 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A039991 (−∞, ∞)
• A257608: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.
  1 · 1, 1 · 1, 20, 1 · 1, 219, 219, 1 … row sums 1, 2, 22, 440, 12760, 484880, 22789360, …
  A257608 (−∞, 0) nonnegative
• A344565: Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
  3 · 15, 15 · 45, 90, 45 · 105, 315, 315, 105 … row sums 3, 30, 180, 840, 3360, 12096, 40320, …
  A344565 (−∞, 0) nonnegative rational roots
• A356546: Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
  1 · 2, 2 · 6, 12, 6 · 20, 60, 60, 20 … row sums 1, 4, 24, 160, 1120, 8064, 59136, …
  A356546 (−∞, 0) nonnegative rational roots
• A142458: Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).
  1 · 1, 1 · 1, 8, 1 · 1, 39, 39, 1 … row sums 1, 2, 10, 80, 880, 12320, 209440, …
  A142458 (−∞, 0) nonnegative
• A225467: Triangle read by rows, T(n, k) = 4^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, 16 · 27, 316, 336, 64 … row sums 1, 7, 65, 743, 9921, 150151, 2526593, …
  A225467 (−∞, 0) nonnegative
• A035469: Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.
  1 · 4, 1 · 28, 12, 1 · 280, 160, 24, 1 … row sums 1, 5, 41, 465, 6721, 117941, 2433145, …
  A035469 (−∞, 0) nonnegative
• A207606: Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.
  4, 7, 2 · 5, 16, 11, 2 · 6, 30, 36, 15, 2 … row sums 13, 34, 89, 233, 610, 1597, 4181, …
  A207606 (−4, 0) nonnegative
• A257620: Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.
  1 · 3, 3 · 9, 36, 9 · 27, 297, 297, 27 … row sums 1, 6, 54, 648, 9720, 174960, 3674160, …
  A257620 (−∞, 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
• A163936: Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).
  1 · 1 · 2, 1 · 6, 8, 1 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  A163936 (−∞, 0) nonnegative
• A038324: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.
  1 · 11, 10 · 121, 220, 100 · 1331, 3630, 3300, 1000 … row sums 1, 21, 441, 9261, 194481, 4084101, …
  A038324 (−∞, 0) nonnegative rational roots
• A257622: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.
  1 · 4, 4 · 16, 56, 16 · 64, 552, 552, 64 … row sums 1, 8, 88, 1232, 20944, 418880, 9634240, …
  A257622 (−∞, 0) nonnegative
• A201637: Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
  1 · 1 · 1, 2 · 1, 8, 6 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  A201637 (−∞, 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
• A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
  1 · 6, 5 · 36, 60, 25 · 216, 540, 450, 125 … row sums 1, 11, 121, 1331, 14641, 161051, …
  A038259 (−∞, 0) nonnegative rational roots
• 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
• A008955: Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.
  1 · 1, 1 · 1, 5, 4 · 1, 14, 49, 36 … row sums 1, 2, 10, 100, 1700, 44200, 1635400, …
  A008955 (−∞, 0) nonnegative rational roots
• A322944: Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.
  1 · 1, 1 · 2, 6, 1 · 6, 38, 15, 1 … row sums 1, 2, 9, 60, 513, 5286, 63417, 865824, …
  A322944 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
  144, 48, 4 · 1728, 864, 144, 8 … row sums 196, 2744, 38416, 537824, 7529536, …
  (−∞, 0) nonnegative rational roots
• 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
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
  144, 240, 100 · 1728, 4320, 3600, 1000 … row sums 484, 10648, 234256, 5153632, 113379904, …
  (−∞, 0) nonnegative rational roots
• A128813: Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).
  1 · 1, 1 · 1, 4, 3 · 1, 10, 27, 18 … row sums 1, 2, 8, 56, 616, 9856, 216832, …
  A128813 (−∞, 0) nonnegative rational roots
• 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
• 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
• A257614: Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*n + 2.
  1 · 2, 2 · 4, 28, 4 · 8, 244, 244, 8 … row sums 1, 4, 36, 504, 9576, 229824, 6664896, …
  A257614 (−∞, 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
• chebyshev_u
  1 · 0, 2 · -1, 0, 4 · 0, -4, 0, 8 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  (−1, 1)
• A013613: Triangle of coefficients in expansion of (1+6x)^n.
  1 · 1, 6 · 1, 12, 36 · 1, 18, 108, 216 … row sums 1, 7, 49, 343, 2401, 16807, 117649, …
  A013613 (−∞, 0) nonnegative rational roots
• A173018: Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
  1 · 1 · 1, 1 · 1, 4, 1 … row sums 1, 1, 2, 6, 24, 120, 720, 5040, …
  A173018 (−∞, 0) nonnegative
• A130534: Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, …
  1 · 1, 1 · 2, 3, 1 · 6, 11, 6, 1 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A130534 (−∞, −1] nonnegative rational roots
• A027466: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).
  1 · 7, 1 · 49, 14, 1 · 343, 147, 21, 1 … row sums 1, 8, 64, 512, 4096, 32768, 262144, …
  A027466 (−∞, 0) nonnegative rational roots
• 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
• A008278: Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
  1 · 1, 1 · 1, 3, 1 · 1, 6, 7, 1 … row sums 1, 2, 5, 15, 52, 203, 877, 4140, …
  A008278 (−∞, 0) nonnegative
• A125662: A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
  1 · 3, 1 · 8, 6, 1 · 21, 25, 9, 1 … row sums 1, 4, 15, 56, 209, 780, 2911, 10864, …
  A125662 (−5, −1) nonnegative
• A257624: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 5.
  1 · 5, 5 · 25, 80, 25 · 125, 915, 915, 125 … row sums 1, 10, 130, 2080, 39520, 869440, …
  A257624 (−∞, 0) nonnegative
• A358735: Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.
  1 · 1, 1 · 1, 4, 2 · 1, 10, 16, 6 … row sums 1, 2, 7, 33, 191, 1304, 10241, 90865, …
  A358735 (−2, 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
• A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
  1 · 12, 9 · 144, 216, 81 · 1728, 3888, 2916, 729 … row sums 1, 21, 441, 9261, 194481, 4084101, …
  A038335 (−∞, 0) nonnegative rational roots
• A391533: The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
  4, 4, 1 · 4, 8, 5, 1 · 4, 12, 13, 6, 1 … row sums 9, 18, 36, 72, 144, 288, 576, 1152, …
  A391533 (−∞, 0) nonnegative rational roots
• 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
• A086645: Triangle read by rows: T(n, k) = binomial(2n, 2k).
  1 · 1, 1 · 1, 6, 1 · 1, 15, 15, 1 … row sums 1, 2, 8, 32, 128, 512, 2048, 8192, …
  A086645 (−∞, 0) nonnegative
• A053122: Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).
  1 · -2, 1 · 3, -4, 1 · -4, 10, -6, 1 … row sums 1, -1, 0, 1, -1, 0, 1, -1, …
  A053122 (0, 4)
• A060920: Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).
  1 · 2, 1 · 5, 5, 1 · 13, 20, 9, 1 … row sums 1, 3, 11, 43, 171, 683, 2731, 10923, …
  A060920 (−∞, −1] nonnegative
• A257618: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 8*x + 2.
  1 · 2, 2 · 4, 40, 4 · 8, 472, 472, 8 … row sums 1, 4, 48, 960, 26880, 967680, 42577920, …
  A257618 (−∞, 0) nonnegative
• Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).
  49, 21, 1 · 343, 343, 42, 1 · 2401, 5145, 1225, 70, 1 … row sums 71, 729, 8842, 125399, 2026249, …
  (−∞, 0) nonnegative