parent_provers

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

135 sequences

• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
  1 · 3, 2 · 9, 12, 4 · 27, 54, 36, 8 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A038220 (−∞, −1] nonnegative rational roots
• A028326: Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).
  2 · 2, 2 · 2, 4, 2 · 2, 6, 6, 2 … row sums 2, 4, 8, 16, 32, 64, 128, 256, …
  A028326 (−∞, 0) nonnegative rational roots
• A034870: Even-numbered rows of Pascal's triangle.
  1 · 1, 2, 1 · 1, 4, 6, 4, 1 · 1, 6, 15, 20, 15, 6, 1 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A034870 (−∞, −1] nonnegative rational roots
• A102547: Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318).
  1 · 1 · 1 · 1, 1 … row sums 1, 1, 1, 2, 3, 4, 6, 9, …
  A102547 (−∞, 0] nonnegative
• A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
  64, 48, 9 · 512, 576, 216, 27 … row sums 121, 1331, 14641, 161051, 1771561, …
  A038281 (−∞, 0) nonnegative rational roots
• A204420: Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.
  1 · 0, 1 · 0, 6, 3 · 0, 120, 90, 15 … row sums 1, 1, 9, 225, 11025, 893025, 108056025, …
  A204420 (−∞, 0] nonnegative rational roots
• basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x -> x + 1
  1 · 0, 1 · 0, 3, 1 · 0, 18, 9, 1 … row sums 1, 1, 4, 28, 280, 3640, 58240, 1106560, …
  (−∞, 0] nonnegative rational roots
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
  1 · 0, 1 · 0, 0, 1 · 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0] [0, ∞) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
  121, 132, 36 · 1331, 2178, 1188, 216 … row sums 289, 4913, 83521, 1419857, 24137569, …
  (−∞, 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+8*x+x^2)/(1-x…
  1, 8, 1 · 1, 19, 19, 1 · 1, 42, 114, 42, 1 … row sums 10, 40, 200, 1200, 8400, 67200, 604800, …
  (−∞, 0) nonnegative
• A038255: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).
  1 · 6, 1 · 36, 12, 1 · 216, 108, 18, 1 … row sums 1, 7, 49, 343, 2401, 16807, 117649, …
  A038255 (−∞, 0) nonnegative rational roots
• A286718: Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.
  1 · 1, 1 · 4, 5, 1 · 28, 39, 12, 1 … row sums 1, 2, 10, 80, 880, 12320, 209440, …
  A286718 (−∞, 0) nonnegative rational roots
• Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
  9, 48, 64 · 27, 216, 576, 512 … row sums 121, 1331, 14641, 161051, 1771561, …
  (−∞, 0) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
  100, 240, 144 · 1000, 3600, 4320, 1728 … row sums 484, 10648, 234256, 5153632, 113379904, …
  (−∞, 0) nonnegative rational roots
• A109692: Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).
  1 · 1, 1 · 1, 4, 3 · 1, 9, 23, 15 … row sums 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  A109692 (−∞, 0) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
  121, 110, 25 · 1331, 1815, 825, 125 … row sums 256, 4096, 65536, 1048576, 16777216, …
  (−∞, 0) nonnegative rational roots
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
  1 · 1, 4 · 1, 8, 16 · 1, 12, 48, 64 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A013611 (−∞, 0) [−1, 0] nonnegative rational roots
• A038322: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.
  1 · 11, 8 · 121, 176, 64 · 1331, 2904, 2112, 512 … row sums 1, 19, 361, 6859, 130321, 2476099, …
  A038322 (−∞, 0) nonnegative rational roots
• A145324: Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....
  1 · 1, 2 · 1, 5, 6 · 1, 9, 26, 24 … row sums 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  A145324 (−∞, 0) [−1, 0] nonnegative rational roots
• A139548: Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.
  1 · 2, 4, 2 · 4, 16, 24, 16, 4 … row sums 1, 8, 64, 512, 4096, 32768, 262144, …
  A139548 (−∞, 0] nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.
  144, 264, 121 · 1728, 4752, 4356, 1331 … row sums 529, 12167, 279841, 6436343, 148035889, …
  (−∞, 0) nonnegative rational roots
• A103633: Triangle read by rows: triangle of repeated stepped binomial coefficients.
  1 · 0, 1 · 0, 1, 1 · 0, 0, 1, 1 … row sums 1, 1, 2, 2, 4, 4, 8, 8, …
  A103633 (−∞, 0] nonnegative rational roots
• A038279: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.
  1 · 8, 1 · 64, 16, 1 · 512, 192, 24, 1 … row sums 1, 9, 81, 729, 6561, 59049, 531441, …
  A038279 (−∞, 0) nonnegative rational roots
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A010054 (−∞, −1] [−1, 0] [−3−2√2, −3+2√2] nonnegative rational roots
• A090582: T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.
  1 · 2, 1 · 6, 6, 1 · 24, 36, 14, 1 … row sums 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  A090582 (−∞, 0] nonnegative
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0
  1 · 0, -1 · 0, 0, 1 · 0, 0, 0, -1 … row sums 1, -1, 1, -1, 1, -1, 1, -1, …
  [−1, 0] rational roots
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
  1 · 4, 1 · 16, 8, 1 · 64, 48, 12, 1 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A038231 (−∞, −1] nonnegative rational roots
• A038291: Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.
  1 · 9, 1 · 81, 18, 1 · 729, 243, 27, 1 … row sums 1, 10, 100, 1000, 10000, 100000, …
  A038291 (−∞, 0) nonnegative rational roots
• A152830: Triangle read by rows, A007318 rows repeated four times .
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 2, 2, 2, 2, …
  A152830 (−∞, 0] nonnegative rational roots
• A152198: Triangle read by rows, A007318 rows repeated.
  1 · 1 · 1, 1 · 1, 1 … row sums 1, 1, 2, 2, 4, 4, 8, 8, …
  A152198 A152815 (−∞, 0] nonnegative rational roots
• A290319: Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.
  1 · 1, 1 · 5, 6, 1 · 45, 59, 15, 1 … row sums 1, 2, 12, 120, 1680, 30240, 665280, …
  A290319 (−∞, 0) nonnegative rational roots
• A257609: 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) = 2*x + 2.
  1 · 2, 2 · 4, 16, 4 · 8, 88, 88, 8 … row sums 1, 4, 24, 192, 1920, 23040, 322560, …
  A257609 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
  100, 60, 9 · 1000, 900, 270, 27 … row sums 169, 2197, 28561, 371293, 4826809, …
  (−∞, 0) nonnegative rational roots
• basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
  1 · 0, -1 · 0, 0, 1 · 0, 0, 0, -1 … row sums 1, -1, 1, -1, 1, -1, 1, -1, …
  [−1, 0] rational roots
• A094638: Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first …
  1 · 1, 1 · 1, 3, 2 · 1, 6, 11, 6 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A094638 (−∞, 0) nonnegative rational roots
• A038323: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.
  1 · 11, 9 · 121, 198, 81 · 1331, 3267, 2673, 729 … row sums 1, 20, 400, 8000, 160000, 3200000, …
  A038323 (−∞, 0) nonnegative rational roots
• A026374: Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 …
  1 · 1, 1 · 1, 3, 1 · 1, 4, 4, 1 … row sums 1, 2, 5, 10, 25, 50, 125, 250, …
  A026374 (−∞, 0] nonnegative
• A028338: Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
  1 · 1, 1 · 3, 4, 1 · 15, 23, 9, 1 … row sums 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  A028338 (−∞, 0) nonnegative rational roots
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.
  3 · 6, 3 · 12, 12, 3 · 24, 36, 18, 3 … row sums 3, 9, 27, 81, 243, 729, 2187, 6561, …
  A120910 (−∞, 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+12*x+x^2)/(1-…
  1, 12, 1 · 1, 27, 27, 1 · 1, 58, 162, 58, 1 … row sums 14, 56, 280, 1680, 11760, 94080, …
  (−∞, 0) nonnegative
• A269944: Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.
  1 · 0, 1 · 0, 1, 1 · 0, 4, 5, 1 … row sums 1, 1, 2, 10, 100, 1700, 44200, 1635400, …
  A269944 (−∞, 0] nonnegative rational roots
• A169803: Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).
  1 · 1, 1 · 1, 2 · 1, 3, 1 … row sums 1, 2, 3, 5, 8, 13, 21, 34, …
  A169803 (−∞, 0] nonnegative
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
  1 · 2, 3 · 4, 12, 9 · 8, 36, 54, 27 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A013620 (−∞, 0) [−1, 0] nonnegative rational roots
• Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.
  1, 3, 2 · 1, 5, 8, 4 · 1, 6, 13, 12, 4 … row sums 6, 18, 36, 108, 216, 648, 1296, 3888, …
  (−∞, 0] nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
  144, 168, 49 · 1728, 3024, 1764, 343 … row sums 361, 6859, 130321, 2476099, 47045881, …
  (−∞, 0) nonnegative rational roots
• basis-map coefficients of A scaled Hermite triangle.
  0, 0, 1 · 0, 0, 0, 1 · 0, 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0] nonnegative rational roots
• A137320: Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows…
  1 · 0, 2 · 0, 2, 4 · 0, 4, 12, 8 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A137320 (−∞, 0] nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
  144, 72, 9 · 1728, 1296, 324, 27 … row sums 225, 3375, 50625, 759375, 11390625, …
  (−∞, 0) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
  144, 192, 64 · 1728, 3456, 2304, 512 … row sums 400, 8000, 160000, 3200000, 64000000, …
  (−∞, 0) nonnegative rational roots
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
  1 · 1, 3 · 1, 6, 9 · 1, 9, 27, 27 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A013610 (−∞, 0) [−1, 0] nonnegative rational roots