Sequences

3,043 sequences

• A274496 → basis A013609
  1 · 2 · 4, 4 · 8, 8 … row sums 1, 2, 8, 16, 64, 128, 512, 1024, …
  (−∞, 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 T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m = 0.
  1, 11, 1 · 1, 111, 30, 1 · 1, 1111, 681, 58, 1 … row sums 13, 143, 1852, 27563, 473725, 9290396, …
  (−∞, 0) nonnegative
• A131689 → D
  1 · 1, 4 · 1, 12, 18 · 1, 28, 108, 96 … row sums 0, 1, 5, 31, 233, 2071, 21305, 249271, …
  (−∞, 0] nonnegative
• A227450 → θ+1
  2 · 2, 8 · 2, 24, 24 · 2, 56, 144, 64 … row sums 2, 10, 50, 266, 1522, 9354, 61490, …
  (−∞, 0] nonnegative
• A196347 → Pₙ + Pₙ₊₁
  2, 1 · 3, 5, 2 · 8, 22, 20, 6 · 30, 114, 162, 102, 24 … row sums 3, 10, 56, 432, 4224, 49920, 691200, …
  (−∞, ∞) 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
• A116412 → Veronese 2 (j=1)
  1 · 6 · 21, 1 · 60, 12 … row sums 0, 1, 6, 22, 72, 235, 774, 2557, …
  (−∞, 0] nonnegative
• A249252 → basis A013609
  1 · 5, 8 · 17, 48, 32 · 61, 248, 320, 128 … row sums 1, 13, 97, 757, 5881, 45709, 355249, …
  (−∞, 0] nonnegative
• A007318 → (1/2+(1/2)n)θ+1+n → reverse
  1 · 3, 2 · 6, 9, 3 · 10, 24, 18, 4 … row sums 1, 5, 18, 56, 160, 432, 1120, 2816, …
  (−∞, 0] nonnegative rational roots
• A038322 → basis A008277
  1 · 19, 8 · 361, 368, 64 · 6859, 12824, 5184, 512 … row sums 1, 27, 793, 25379, 883057, 33250795, …
  (−∞, 0] nonnegative
• A143495 → (θ+1)²
  1 · 3, 4 · 9, 28, 9 · 27, 148, 108, 16 … row sums 1, 7, 46, 299, 1967, 13248, 91903, …
  (−∞, 0] nonnegative
• A103633 → (θ+1)²
  1 · 0, 4 · 0, 4, 9 · 0, 0, 9, 16 … row sums 1, 4, 13, 25, 66, 102, 248, 344, …
  (−∞, 0] nonnegative
• A257612: 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) = 4*x + 2.
  1 · 2, 2 · 4, 24, 4 · 8, 184, 184, 8 … row sums 1, 4, 32, 384, 6144, 122880, 2949120, …
  A257612 (−∞, 0) nonnegative
• A039755 → (θ+1)²
  1 · 1, 4 · 1, 16, 9 · 1, 52, 81, 16 … row sums 1, 5, 26, 150, 964, 6836, 52936, …
  (−∞, 0] nonnegative
• A163932: Triangle related to the asymptotic expansion of E(x,m=3,n).
  A130534 → D²∘x (shift +1)
  1 · 3, 3 · 11, 18, 6 · 50, 105, 60, 10 … row sums 1, 6, 35, 225, 1624, 13132, 118124, …
  A163932 (−∞, 0] nonnegative
• A039755 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 24, 34, 12, 1 … row sums 1, 3, 13, 71, 457, 3355, 27509, 248127, …
  (−∞, −1] nonnegative
• A116412 → D
  1 · 6, 2 · 21, 18, 3 · 60, 90, 36, 4 … row sums 0, 1, 8, 42, 190, 797, 3192, 12391, …
  (−∞, 0] nonnegative
• A135838 → (θ+1)²
  1 · 2, 8 · 2, 16, 18 · 4, 48, 108, 64 … row sums 1, 10, 36, 224, 640, 3456, 8960, 45056, …
  (−∞, 0] nonnegative
• A131689: Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.
  1 · 0, 1 · 0, 1, 2 · 0, 1, 6, 6 … row sums 1, 1, 3, 13, 75, 541, 4683, 47293, …
  A131689 (−1, 0] nonnegative
• A007318 → (1/3+(1/2)n+(1/6)n²)θ+1+(3/2)n+(1/2)n² → reverse
  1 · 4, 3 · 10, 16, 6 · 20, 50, 40, 10 … row sums 1, 7, 32, 120, 400, 1232, 3584, 9984, …
  (−∞, 0] nonnegative rational roots
• A272866 → basis A007318
  1 · 5, 5, 1 · 25, 50, 35, 10, 1 … row sums 1, 11, 121, 1331, 14641, 161051, …
  (−∞, −1] nonnegative
• basis-map coefficients of A021012 → (θ+1)²
  1 · 0, -4 · 0, 0, 18 · 0, 0, 0, -96 … row sums 1, -4, 18, -96, 600, -4320, 35280, …
  (−∞, 0] rational roots
• A144445: Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 23, 1 · 1, 206, 206, 1 … row sums 1, 2, 25, 414, 9697, 293808, 10938775, …
  A144445 (−∞, 0) nonnegative
• basis-map coefficients of A026729 → basis A056242 → θ+1
  1 · 1, 4 · 1, 10, 12 · 1, 18, 48, 32 … row sums 1, 5, 23, 99, 408, 1632, 6388, 24596, …
  (−∞, 0] nonnegative
• A023531 → (θ+1)² → (θ+1)²
  1 · 0, 16 · 0, 0, 81 · 0, 0, 0, 256 … row sums 1, 16, 81, 256, 625, 1296, 2401, 4096, …
  (−∞, 0] nonnegative rational roots
• A204420 → basis A007318
  1 · 1, 1 · 9, 12, 3 · 225, 345, 135, 15 … row sums 1, 2, 24, 720, 40320, 3628800, …
  (−∞, 0] nonnegative rational roots
• A038338 → basis A007318
  576, 576, 144 · 13824, 20736, 10368, 1728 … row sums 1296, 46656, 1679616, 60466176, …
  (−∞, 0] nonnegative rational roots
• A008288 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 10, 15, 4 … row sums 1, 3, 10, 30, 87, 245, 676, 1836, …
  (−∞, 0] nonnegative
• A207607 → D
  2 · 5, 4 · 9, 18, 6 · 14, 50, 39, 8 … row sums 0, 2, 9, 33, 111, 355, 1098, 3316, …
  (−∞, 0] nonnegative
• A053122 → x ↦ x+1
  1 · -1, 1 · 0, -2, 1 · 1, 1, -3, 1 … row sums 1, 0, -1, 0, 1, 0, -1, 0, …
  (−1, 3)
• basis-map coefficients of A371080, shifted → x ↦ x−1
  1 · -1, 1 · -2, 1, 1 · -10, 3, 6, 1 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  (−∞, 1] 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
• A038226 → basis A007318
  121, 176, 64 · 1331, 2904, 2112, 512 … row sums 361, 6859, 130321, 2476099, 47045881, …
  (−∞, 0] nonnegative rational roots
• A062196 → (θ+1)²
  1 · 1, 12 · 1, 32, 54 · 1, 60, 270, 160 … row sums 1, 13, 87, 491, 2562, 12762, 61611, …
  (−∞, 0] nonnegative
• A081578 → Veronese 2
  1 · 1 · 1, 1 · 1, 9 … row sums 1, 1, 2, 10, 35, 91, 260, 820, …
  (−∞, 0] nonnegative
• A007318 → basis A049020 → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 9, 22, 15 … row sums 1, 3, 11, 47, 227, 1215, 7107, 44959, …
  (−∞, 0] nonnegative
• A007318 → Veronese 2 (j=1)
  1 · 2 · 3, 1 · 4, 4 … row sums 0, 1, 2, 4, 8, 16, 32, 64, …
  (−∞, 0] nonnegative
• A081581 → basis A013610
  1 · 2, 3 · 10, 30, 9 · 32, 144, 162, 27 … row sums 1, 5, 49, 365, 3001, 23765, 190849, …
  (−∞, 0] nonnegative
• A131689 → D+1
  1 · 1, 1 · 1, 5, 2 · 1, 13, 24, 6 … row sums 1, 2, 8, 44, 308, 2612, 25988, 296564, …
  (−∞, 0] nonnegative
• A013620 → D+1
  1 · 5, 3 · 16, 30, 9 · 44, 144, 135, 27 … row sums 1, 8, 55, 350, 2125, 12500, 71875, …
  (−∞, 0] nonnegative rational roots
• A010054 → reverse → D+1
  1 · 3, 3 · 0, 12, 6 · 0, 0, 30, 10 … row sums 1, 6, 18, 40, 75, 126, 196, 288, …
  (−∞, 0] nonnegative rational roots
• A099093 → x ↦ x−12
  1 · -36, 3 · 1260, -213, 9 · -44064, 11232, -954, 27 … row sums 1, -33, 1056, -33759, 1079199, …
  (32/3, 12]
• A340556: E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.
  A163936 → reverse
  1 · 0, 1 · 0, 1, 2 · 0, 1, 8, 6 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  A340556 (−∞, 0] nonnegative
• A143685 → basis A007318
  1 · 2, 1 · 13, 13, 1 · 44, 66, 24, 1 … row sums 1, 3, 27, 135, 891, 5103, 31347, …
  (−∞, −1] nonnegative
• 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
• A133080 → Veronese 2 (j=1)
  1 · 0, 1 · 0, 0, 1 · 0, 0, 0, 1 … row sums 0, 1, 0, 1, 0, 1, 0, 1, …
  (−∞, 0] nonnegative rational roots
• A053122 → D+1
  1 · -1, 1 · -1, -2, 1 · 6, -2, -3, 1 … row sums 1, 0, -2, 2, 1, -4, 3, 2, …
  (−∞, ∞)
• A079638 → basis A013610
  1 · 4, 3 · 24, 33, 9 · 192, 366, 189, 27 … row sums 1, 7, 66, 774, 10800, 174240, 3185280, …
  (−∞, 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