parent_provers / periodic_product

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

85 sequences

• 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
• 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
• 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
• 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
• A007318: Binomial
  1 · 1, 1 · 1, 2, 1 · 1, 3, 3, 1 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A007318 (−∞, −1] nonnegative rational roots
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0
  1 · 0, 0, 1 · 0, 0, 0, 0, 1 · 0, 0, 0, 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)*9^(i-j)*11^j.
  81, 198, 121 · 729, 2673, 3267, 1331 … row sums 400, 8000, 160000, 3200000, 64000000, …
  (−∞, 0) nonnegative rational roots
• A038208: Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
  1 · 2, 2 · 4, 8, 4 · 8, 24, 24, 8 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A038208 (−∞, −1] nonnegative rational roots
• A272867: Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
  1 · 1, 4, 1 · 1, 8, 18, 8, 1 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …
  A272867 (−∞, 0] nonnegative
• A133087: A133080 * A007318.
  1 · 2, 1 · 1, 2, 1 · 2, 5, 4, 1 … row sums 1, 3, 4, 12, 16, 48, 64, 192, …
  A133087 (−∞, −1] nonnegative rational roots
• 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
• Riordan array ((1+x)^3,x).
  1, 3, 3, 1 · 0, 1, 3, 3, 1 · 0, 0, 1, 3, 3, 1 … row sums 8, 8, 8, 8, 8, 8, 8, 8, …
  (−∞, 0] nonnegative rational roots
• 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
• A133080: Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
  1 · 1, 1 · 0, 0, 1 · 0, 0, 1, 1 … row sums 1, 2, 1, 2, 1, 2, 1, 2, …
  A133080 (−∞, 0] nonnegative rational roots
• 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
• 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
• A200536: Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.
  1 · 1, 3, 2 · 1, 6, 13, 12, 4 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …
  A200536 (−∞, 0] nonnegative rational roots
• 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
• 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
• 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
• A027467: Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).
  1 · 15, 1 · 225, 30, 1 · 3375, 675, 45, 1 … row sums 1, 16, 256, 4096, 65536, 1048576, …
  A027467 (−∞, 0) nonnegative rational roots