parent_provers

Shared parent provers on the node's own rule. All operations

135 sequences

• Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-…
  1, 10, 1 · 1, 23, 23, 1 · 1, 50, 138, 50, 1 … row sums 12, 48, 240, 1440, 10080, 80640, …
  (−∞, 0) nonnegative
• A118931: Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3*k>=0.
  1 · 1 · 1 · 1, 2 … row sums 1, 1, 1, 3, 9, 21, 81, 351, …
  A118931 (−∞, 0] nonnegative
• A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
  1 · 0, 1 · 0, 0, 1 · 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A023531 [−1, 0] [0, ∞) nonnegative rational roots
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
  1 · 1, 2 · 1, 4, 4 · 1, 6, 12, 8 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A013609 (−∞, 0) [−1, 0] nonnegative rational roots
• A152842: Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
  1 · 1, 3 · 1, 4, 3 · 1, 7, 15, 9 … row sums 1, 4, 8, 32, 64, 256, 512, 2048, …
  A152842 (−∞, 0] nonnegative rational roots
• basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
  1 · -1, -1 · 1, 2, 1 · -1, -3, -3, -1 … row sums 1, -2, 4, -8, 16, -32, 64, -128, …
  (−∞, −1] rational roots
• A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
  1 · 2, 2 · 6, 12, 6 · 24, 72, 72, 24 … row sums 1, 4, 24, 192, 1920, 23040, 322560, …
  A156992 (−∞, 0) nonnegative rational roots
• A011973: Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.
  1 · 1 · 1, 1 · 1, 2 … row sums 1, 1, 2, 3, 5, 8, 13, 21, …
  A011973 (−∞, 0] nonnegative
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).
  3 · 3, 6 · 3, 12, 12 · 3, 18, 36, 24 … row sums 3, 9, 27, 81, 243, 729, 2187, 6561, …
  A120909 (−∞, 0) nonnegative rational roots
• Riordan array (1-x,x/(1-x)).
  0, 0, 1 · 0, 0, 1, 1 · 0, 0, 1, 2, 1 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, 0] nonnegative rational roots
• A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
  1 · 5, 1 · 25, 10, 1 · 125, 75, 15, 1 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …
  A038243 (−∞, −1] nonnegative rational roots
• A274498: Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).
  1 · 3 · 3, 6 · 9, 18 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A274498 (−∞, 0] nonnegative rational roots
• A161198: Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).
  1 · 1, 2 · 3, 8, 4 · 15, 46, 36, 8 … row sums 1, 3, 15, 105, 945, 10395, 135135, …
  A161198 (−∞, 0) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
  121, 154, 49 · 1331, 2541, 1617, 343 … row sums 324, 5832, 104976, 1889568, 34012224, …
  (−∞, 0) nonnegative rational roots
• A299989: Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) for n >= k > 0.
  0, 1 · 0, 3, 4, 1 · 0, 9, 24, 22, 8, 1 … row sums 1, 8, 64, 512, 4096, 32768, 262144, …
  A299989 (−∞, 0] nonnegative rational roots
• A059364: Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.
  1 · 2, 1 · 6, 7, 2 · 24, 46, 29, 6 … row sums 1, 3, 15, 105, 945, 10395, 135135, …
  A059364 (−∞, −1] nonnegative rational roots
• A038325: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.
  1 · 11, 11 · 121, 242, 121 · 1331, 3993, 3993, 1331 … row sums 1, 22, 484, 10648, 234256, 5153632, …
  A038325 (−∞, 0) nonnegative rational roots
• A038327: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.
  1 · 12, 1 · 144, 24, 1 · 1728, 432, 36, 1 … row sums 1, 13, 169, 2197, 28561, 371293, …
  A038327 (−∞, 0) nonnegative rational roots
• A225470: Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
  1 · 2, 1 · 10, 7, 1 · 80, 66, 15, 1 … row sums 1, 3, 18, 162, 1944, 29160, 524880, …
  A225470 (−∞, 0) nonnegative rational roots
• A088996: Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.
  1 · 0, 1 · 0, 1, 2 · 0, 2, 7, 6 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  A088996 [−1, 0] nonnegative rational roots
• A027465: Cube of lower triangular normalized binomial matrix.
  1 · 3, 1 · 9, 6, 1 · 27, 27, 9, 1 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A027465 (−∞, −1] nonnegative rational roots
• A038221: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
  1 · 3, 3 · 9, 18, 9 · 27, 81, 81, 27 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …
  A038221 (−∞, 0) nonnegative rational roots
• A136124: Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).
  1 · 2, 1 · 6, 5, 1 · 24, 26, 9, 1 … row sums 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  A136124 (−∞, −1] nonnegative rational roots
• A100861: Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
  1 · 1 · 1, 1 · 1, 3 … row sums 1, 1, 2, 4, 10, 26, 76, 232, …
  A100861 A144299 (−∞, 0] nonnegative
• A152828: Triangle read by rows, A007318 rows repeated three times.
  1 · 1 · 1 · 1, 1 … row sums 1, 1, 1, 2, 2, 2, 4, 4, …
  A152828 (−∞, 0] nonnegative rational roots
• Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x…
  1, 6, 1 · 1, 15, 15, 1 · 1, 34, 90, 34, 1 … row sums 8, 32, 160, 960, 6720, 53760, 483840, …
  (−∞, 0) nonnegative
• A225471: Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
  1 · 3, 1 · 21, 10, 1 · 231, 131, 21, 1 … row sums 1, 4, 32, 384, 6144, 122880, 2949120, …
  A225471 (−∞, 0) nonnegative rational roots
• A038326: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.
  1 · 11, 12 · 121, 264, 144 · 1331, 4356, 4752, 1728 … row sums 1, 23, 529, 12167, 279841, 6436343, …
  A038326 (−∞, 0) nonnegative rational roots
• A318772: Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 4, 7, 10, 13, …
  A318772 (−∞, 0] nonnegative
• A317496: Triangle T(n,k) = T(n-1,k) + 3*T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.
  1 · 1 · 1 · 1, 3 … row sums 1, 1, 1, 4, 7, 10, 22, 43, …
  A317496 (−∞, 0] nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
  144, 120, 25 · 1728, 2160, 900, 125 … row sums 289, 4913, 83521, 1419857, 24137569, …
  (−∞, 0) nonnegative rational roots
• A135838: Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
  1 · 2, 2 · 2, 4, 2 · 4, 12, 12, 4 … row sums 1, 4, 8, 32, 64, 256, 512, 2048, …
  A135838 (−∞, 0] nonnegative rational roots
• A147716: Triangle of coefficients in expansion of (14 + x)^n.
  1 · 14, 1 · 196, 28, 1 · 2744, 588, 42, 1 … row sums 1, 15, 225, 3375, 50625, 759375, …
  A147716 (−∞, 0) nonnegative rational roots
• A038271: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.
  1 · 7, 5 · 49, 70, 25 · 343, 735, 525, 125 … row sums 1, 12, 144, 1728, 20736, 248832, …
  A038271 (−∞, 0) nonnegative rational roots
• A038303: Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.
  1 · 10, 1 · 100, 20, 1 · 1000, 300, 30, 1 … row sums 1, 11, 121, 1331, 14641, 161051, …
  A038303 (−∞, 0) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
  144, 96, 16 · 1728, 1728, 576, 64 … row sums 256, 4096, 65536, 1048576, 16777216, …
  (−∞, 0) nonnegative rational roots
• A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).
  1 · 0, 2 · 0, 0, 4 · 0, 0, 0, 8 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A130123 [−1, 0] nonnegative rational roots
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
  1 · 2, 1 · 4, 4, 1 · 8, 12, 6, 1 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A038207 (−∞, −1] nonnegative rational roots
• A118394: Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.
  1 · 1 · 1 · 1, 6 … row sums 1, 1, 1, 7, 25, 61, 481, 2731, …
  A118394 (−∞, 0] nonnegative
• Triangle of coefficients in expansion of (1+11x)^n.
  1, 22, 121 · 1, 33, 363, 1331 … row sums 144, 1728, 20736, 248832, 2985984, …
  (−∞, 0) nonnegative rational roots
• A272866: Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
  1 · 1, 3, 1 · 1, 6, 11, 6, 1 · 1, 9, 30, 45, 30, 9, 1 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A272866 (−∞, 0] nonnegative
• A274496: Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).
  1 · 2 · 2, 2 · 4, 4 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A274496 (−∞, 0] nonnegative rational roots
• A038277: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.
  1 · 7, 11 · 49, 154, 121 · 343, 1617, 2541, 1331 … row sums 1, 18, 324, 5832, 104976, 1889568, …
  A038277 (−∞, 0) nonnegative rational roots
• A132393: Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
  1 · 0, 1 · 0, 1, 1 · 0, 2, 3, 1 … row sums 1, 1, 2, 6, 24, 120, 720, 5040, …
  A132393 (−∞, 0] nonnegative rational roots
• A188440: Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.
  1 · 1 · 1, 2 · 1, 2 … row sums 1, 1, 3, 3, 9, 9, 27, 27, …
  A188440 (−∞, 0] nonnegative rational roots
• Triangle of coefficients in expansion of (1+5x)^n.
  1, 10, 25 · 1, 15, 75, 125 · 1, 20, 150, 500, 625 … row sums 36, 216, 1296, 7776, 46656, 279936, …
  (−∞, 0) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
  64, 96, 36 · 512, 1152, 864, 216 … row sums 196, 2744, 38416, 537824, 7529536, …
  (−∞, 0) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
  144, 288, 144 · 1728, 5184, 5184, 1728 … row sums 576, 13824, 331776, 7962624, 191102976, …
  (−∞, 0) nonnegative rational roots
• Unsigned 2-Stirling numbers of the first kind.
  6, 5, 1 · 24, 26, 9, 1 · 120, 154, 71, 14, 1 … row sums 12, 60, 360, 2520, 20160, 181440, …
  (−∞, 0) nonnegative rational roots
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
  1 · 3, 5 · 9, 30, 25 · 27, 135, 225, 125 … row sums 1, 8, 64, 512, 4096, 32768, 262144, …
  A013622 (−∞, 0) [−1, 0] nonnegative rational roots