theorem_check

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

304 sequences

• Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).
  4, 16, 8 · 8, 72, 144, 48 · 16, 256, 1152, 1536, 384 … row sums 28, 272, 3344, 49472, 852928, 16758016, …
  (−∞, 0) nonnegative
• 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
• A207823: Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).
  1 · 4, 1 · 15, 8, 1 · 56, 46, 12, 1 … row sums 1, 5, 24, 115, 551, 2640, 12649, 60605, …
  A207823 (−6, −2) nonnegative
• A282629: Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].
  1 · 1, 3 · 1, 15, 9 · 1, 63, 108, 27 … row sums 1, 4, 25, 199, 1876, 20257, 245017, …
  A282629 (−∞, 0) nonnegative
• A111125: Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
  1 · 3, 1 · 5, 5, 1 · 7, 14, 7, 1 … row sums 1, 4, 11, 29, 76, 199, 521, 1364, …
  A111125 (−4, 0) nonnegative
• A094816: Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.
  1 · 1, 1 · 1, 3, 1 · 1, 8, 6, 1 … row sums 1, 2, 5, 16, 65, 326, 1957, 13700, …
  A094816 (−∞, 0) nonnegative
• 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
• A207607: Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.
  1 · 1, 2 · 1, 5, 2 · 1, 9, 9, 2 … row sums 1, 3, 8, 21, 55, 144, 377, 987, …
  A207607 (−4, 0) nonnegative
• A144699: Triangle of 5-Eulerian numbers.
  1 · 1, 5 · 1, 16, 25 · 1, 39, 171, 125 … row sums 1, 6, 42, 336, 3024, 30240, 332640, …
  A144699 (−∞, 0) nonnegative
• A280580: Triangle read by rows: T(n,k) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=n.
  1 · 1, 1 · 2, 6, 1 · 5, 30, 15, 1 … row sums 1, 2, 9, 51, 323, 2188, 15511, 113634, …
  A280580 (−∞, 0) nonnegative
• 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
• A026729: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
  1 · 0, 1 · 0, 1, 1 · 0, 0, 2, 1 … row sums 1, 1, 2, 3, 5, 8, 13, 21, …
  A026729 (−4, 0] nonnegative
• A053124: Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in increasing order).
  1 · -2, 4 · 3, -16, 16 · -4, 40, -96, 64 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  A053124 (0, 1)
• basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
  1 · 0, 2 · -3, 3, 3 · 0, -12, 12, 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 1]
• A142461: Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 6.
  1 · 1, 1 · 1, 14, 1 · 1, 111, 111, 1 … row sums 1, 2, 16, 224, 4480, 116480, 3727360, …
  A142461 (−∞, 0) nonnegative
• A008277: Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
  1 · 1, 1 · 1, 3, 1 · 1, 7, 6, 1 … row sums 1, 2, 5, 15, 52, 203, 877, 4140, …
  A008277 (−∞, 0) nonnegative
• 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
• A257615: 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 + 5.
  1 · 5, 5 · 25, 70, 25 · 125, 715, 715, 125 … row sums 1, 10, 120, 1680, 26880, 483840, …
  A257615 (−∞, 0) nonnegative
• A134991: Triangle of Ward numbers T(n,k) read by rows.
  1 · 1, 3 · 1, 10, 15 · 1, 25, 105, 105 … row sums 1, 4, 26, 236, 2752, 39208, 660032, …
  A134991 (−1, 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
• A257623: Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.
  1 · 3, 3 · 9, 48, 9 · 27, 501, 501, 27 … row sums 1, 6, 66, 1056, 22176, 576576, …
  A257623 (−∞, 0) nonnegative
• 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
• A142075: Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.
  1 · 1, 2 · 1, 8, 4 · 1, 22, 44, 8 … row sums 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  A142075 (−∞, 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
• A257625: 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) = 6*n + 3.
  1 · 3, 3 · 9, 54, 9 · 27, 621, 621, 27 … row sums 1, 6, 72, 1296, 31104, 933120, …
  A257625 (−∞, 0) nonnegative
• A convolution triangle of numbers obtained from A036224.
  336, 42, 1 · 4536, 1113, 63, 1 … row sums 379, 5713, 80032, 1080325, 14354557, …
  [−36, 0) nonnegative
• A063007: T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
  1 · 1, 2 · 1, 6, 6 · 1, 12, 30, 20 … row sums 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  A063007 (−1, 0) nonnegative
• A144084: T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.
  1 · 1, 1 · 1, 4, 2 · 1, 9, 18, 6 … row sums 1, 2, 7, 34, 209, 1546, 13327, 130922, …
  A144084 (−∞, 0) nonnegative
• A156365: T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) = 1.
  1, 8, 4 · 1, 22, 44, 8 · 1, 52, 264, 208, 16 … row sums 13, 75, 541, 4683, 47293, 545835, …
  A156365 (−∞, 0) nonnegative
• A179900: Triangle T(n,k) read by rows: coefficient of [x^k] of the polynomial p_n(x)=(5-x)*p_{n-1}(x)-p_{n-2}(x), p_0=1, p_1=5-x.
  1 · 5, -1 · 24, -10, 1 · 115, -73, 15, -1 … row sums 1, 4, 15, 56, 209, 780, 2911, 10864, …
  A179900 (3, 7)
• A137338: Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), ordered by rising powers.
  1 · -1, 1 · 0, -3, 1 · 3, 6, -6, 1 … row sums 1, 0, -2, 4, -4, -4, 44, -236, …
  A137338 (−∞, ∞)
• A285061: Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).
  1 · 1, 4 · 1, 24, 16 · 1, 124, 240, 64 … row sums 1, 5, 41, 429, 5329, 75989, 1215481, …
  A285061 (−∞, 0) nonnegative
• A144697: Triangle of 3-Eulerian numbers.
  1 · 1, 3 · 1, 10, 9 · 1, 25, 67, 27 … row sums 1, 4, 20, 120, 840, 6720, 60480, …
  A144697 (−∞, 0) nonnegative
• A124038: Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.
  1 · -2, 1 · -1, -2, 1 · 2, -2, -2, 1 … row sums 1, -1, -2, -1, 1, 2, 1, -1, …
  A124038 (−2, 3)
• A062196: Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).
  1 · 1, 3 · 1, 8, 6 · 1, 15, 30, 10 … row sums 1, 4, 15, 56, 210, 792, 3003, 11440, …
  A062196 (−∞, 0) nonnegative
• 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
• A106800: Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
  1 · 1 · 1, 1 · 1, 3, 1 … row sums 1, 1, 2, 5, 15, 52, 203, 877, …
  A106800 (−∞, 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
• A055356: Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.
  1 · 1 · 1, 1 · 1, 4, 2 … row sums 1, 1, 2, 7, 36, 245, 2076, 21059, …
  A055356 (−∞, 0) nonnegative
• 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
• Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.
  -2, 0, 1 · 0, -8, 0, 1 · 24, 0, -20, 0, 1 … row sums -1, -7, 5, 145, -5, -6095, -5815, …
  (−∞, ∞)
• A049374: A triangle of numbers related to triangle A030527.
  1 · 6, 1 · 42, 18, 1 · 336, 276, 36, 1 … row sums 1, 7, 61, 649, 8245, 122215, 2069425, …
  A049374 (−∞, 0) nonnegative
• A185410: A decomposition of the double factorials A001147.
  1 · 1 · 1, 2 · 1, 10, 4 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  A185410 (−∞, 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
• A126216: Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A126216 (−∞, −1] nonnegative