Sequences

3,043 sequences

• A395454: Triangle read by rows: T(n, k) = Sum_{j=0..n} C(-j-1, -n-1)*E2(j, n-k), E2 the Eulerian numbers A201637.
  1 · 0, 2 · 0, 6, 2 · 0, 24, 34, 2 … row sums 1, 0, 2, 8, 60, 544, 6040, 79008, …
  A395454 (−∞, 0] nonnegative
• A075497 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 14, 11 · 8, 52, 92, 49 … row sums 1, 5, 29, 201, 1657, 15821, 170389, …
  [−1, 0] nonnegative
• basis-map coefficients of A026729 → basis A056242 → x ↦ x+1
  1 · 3, 2 · 10, 13, 4 · 34, 65, 40, 8 … row sums 1, 5, 27, 147, 801, 4365, 23787, …
  (−∞, −1] nonnegative
• A227450 → x ↦ x/(x+1)
  2 · 2, 6 · 2, 16, 22 · 2, 34, 110, 94 … row sums 2, 8, 40, 240, 1664, 12992, 112256, …
  [−1, 0] nonnegative
• A007318 → (1/3+(1/2)n+(1/6)n²)θ+1+(3/2)n+(1/2)n² → D
  4 · 16, 20 · 40, 100, 60 · 80, 300, 360, 140 … row sums 0, 4, 36, 200, 880, 3360, 11648, 37632, …
  (−∞, 0] nonnegative rational roots
• A172431 → (θ+1)²
  1 · 1, 8 · 1, 16, 27 · 1, 24, 90, 64 … row sums 1, 9, 44, 179, 667, 2352, 7973, 26229, …
  (−∞, 0] nonnegative
• A030528 → x ↦ x−1
  1 · 0, 1 · -1, 0, 1 · 1, -2, 0, 1 … row sums 1, 1, 0, 0, 0, 0, 0, 0, …
  (−3, 1]
• A062190 → θ+1
  1 · 1, 12 · 1, 28, 63 · 1, 48, 252, 224 … row sums 1, 13, 92, 525, 2695, 13013, 60424, …
  (−∞, 0] nonnegative
• A102547 → basis A013609
  1 · 1 · 1 · 2, 2 … row sums 1, 1, 1, 4, 7, 10, 22, 43, …
  (−∞, 0] nonnegative
• A173018 → θ+1
  1 · 1 · 1, 2 · 1, 8, 3 … row sums 1, 1, 3, 12, 60, 360, 2520, 20160, …
  (−∞, 0] nonnegative
• A056242: Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).
  basis-map coefficients of A026729 → basis A056242
  1 · 1, 2 · 1, 5, 4 · 1, 9, 16, 8 … row sums 1, 3, 10, 34, 116, 396, 1352, 4616, …
  A056242 [−1, 0] nonnegative
• A064861 → basis A013611
  6, 28, 32 · 18, 132, 320, 256 … row sums 66, 726, 4356, 47916, 287496, 3162456, …
  (−∞, 0] nonnegative rational roots
• A063007 → basis A008277
  1 · 3, 2 · 13, 24, 6 · 63, 242, 150, 20 … row sums 1, 5, 43, 475, 6231, 93427, 1564249, …
  (−∞, 0] nonnegative
• A039756: Triangle of B-analogs of Stirling numbers of 2nd kind.
  1 · 1, 1 · 1, 4, 1 · 1, 9, 13, 1 … row sums 1, 2, 6, 24, 116, 648, 4088, 28640, …
  A039756 (−∞, 0) nonnegative
• A105070: T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.
  A011973 → x ↦ (1/2)x−1/4
  1 · 2 · 3, 2 · 4, 8 … row sums 1, 2, 5, 12, 29, 70, 169, 408, …
  A105070 (−∞, 1/2] nonnegative
• A144438: Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 5, 1 · 1, 14, 14, 1 … row sums 1, 2, 7, 30, 157, 972, 6961, 56660, …
  A144438 (−∞, 0) nonnegative
• A091042 → basis A007318
  1 · 4, 3 · 16, 20, 5 · 64, 112, 56, 7 … row sums 1, 7, 41, 239, 1393, 8119, 47321, …
  (−∞, −1] nonnegative
• A111577 → Pₙ + Pₙ₊₁
  2, 1 · 2, 6, 1 · 2, 26, 13, 1 · 2, 106, 117, 23, 1 … row sums 3, 9, 42, 249, 1737, 13872, 124401, …
  (−∞, ∞) nonnegative
• A081581 → D
  1 · 8, 2 · 15, 30, 3 · 22, 156, 66, 4 … row sums 0, 1, 10, 48, 248, 1100, 4872, 20608, …
  (−∞, 0] nonnegative
• A102547 → reverse → D
  1 · 0, 2 · 0, 0, 3 · 0, 0, 3, 4 … row sums 1, 2, 3, 7, 13, 21, 36, 61, …
  (−∞, 0] nonnegative
• A008277 → θ+2 (shift +1) → x ↦ x/(x+1)
  2 · 2, 5 · 2, 13, 15 · 2, 27, 72, 52 … row sums 2, 7, 30, 153, 898, 5919, 43102, …
  [−1, 0] nonnegative
• A274498 → basis A013611
  1 · 3 · 9, 24 · 27, 72 … row sums 1, 3, 33, 99, 1089, 3267, 35937, …
  (−∞, 0] nonnegative rational roots
• narayana → D
  1 · 3, 2 · 6, 12, 3 · 10, 40, 30, 4 … row sums 0, 1, 5, 21, 84, 330, 1287, 5005, …
  (−∞, 0] nonnegative
• A075497 → D
  1 · 6, 2 · 28, 24, 3 · 120, 200, 60, 4 … row sums 0, 1, 8, 55, 384, 2841, 22504, 190863, …
  (−∞, 0] nonnegative
• A008277 → θ+1 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 8, 10 · 1, 17, 49, 37 … row sums 1, 4, 19, 104, 641, 4380, 32803, …
  [−1, 0] nonnegative
• A232535 → D
  2 · 8, 6 · 18, 50, 12 · 32, 196, 168, 20 … row sums 0, 2, 14, 80, 416, 2048, 9728, 45056, …
  (−∞, 0] nonnegative
• A249252 → basis A008277
  1 · 5, 4 · 17, 32, 8 · 61, 220, 128, 16 … row sums 1, 9, 57, 425, 3497, 31545, 308697, …
  (−∞, 0] nonnegative
• A207606 → basis A013609
  13, 22, 8 · 34, 88, 68, 16 · 89, 310, 372, 184, 32 … row sums 43, 206, 987, 4729, 22658, 108561, …
  (−∞, 0] nonnegative
• 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
• A063967 → x ↦ x−9
  1 · -8, 1 · 56, -15, 1 · -384, 160, -22, 1 … row sums 1, -7, 42, -245, 1421, -8232, 47677, …
  (4, 8]
• A178822 → Veronese 2 (j=1)
  6 · 42 · 168, 56 · 504, 504 … row sums 0, 6, 42, 224, 1008, 4032, 14784, …
  (−∞, 0] nonnegative
• A135838 → basis A008277
  1 · 4, 2 · 8, 10, 2 · 32, 76, 36, 4 … row sums 1, 6, 20, 148, 604, 5392, 26104, …
  (−∞, 0] nonnegative
• A038271 → Pₙ + Pₙ₊₁
  8, 5 · 56, 75, 25 · 392, 805, 550, 125 … row sums 13, 156, 1872, 22464, 269568, 3234816, …
  (−∞, ∞) nonnegative rational roots
• narayana
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … row sums 1, 2, 5, 14, 42, 132, 429, 1430, …
  (−∞, 0) nonnegative
• delannoy → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 12, 18, 8, 1 … row sums 1, 3, 11, 39, 139, 495, 1763, 6279, …
  (−∞, −1] nonnegative
• A207606 → D
  7, 4 · 16, 22, 6 · 30, 72, 45, 8 … row sums 11, 44, 155, 510, 1608, 4924, 14761, …
  (−∞, 0] nonnegative
• A154986 → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 22, 33, 4 … row sums 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  (−∞, 0] nonnegative
• A038314 → basis A013609
  484, 1056, 576 · 10648, 34848, 38016, 13824 … row sums 2116, 97336, 4477456, 205962976, …
  (−∞, 0] nonnegative rational roots
• A116412 → basis A013611
  1 · 4, 4 · 13, 32, 16 · 43, 168, 192, 64 … row sums 1, 8, 61, 467, 3574, 27353, 209341, …
  (−∞, 0] nonnegative
• A038314 → Pₙ + Pₙ₊₁
  1100, 3840, 4464, 1728 … row sums 11132, 244904, 5387888, 118533536, …
  (−∞, ∞) nonnegative rational roots
• A079638 → basis A013611
  1 · 4, 4 · 24, 44, 16 · 192, 488, 336, 64 … row sums 1, 8, 84, 1080, 16344, 283680, 5544000, …
  (−∞, 0] nonnegative
• A269947 → basis A013611
  1 · 1, 4 · 2, 12, 16 · 18, 116, 192, 64 … row sums 1, 5, 30, 390, 12480, 861120, …
  (−∞, 0] nonnegative rational roots
• A116414 → Veronese 2
  1 · 4 · 13, 1 · 40, 12 … row sums 1, 4, 14, 52, 209, 880, 3773, 16252, …
  (−∞, 0] nonnegative
• A028326 → Pₙ + Pₙ₊₁
  4, 2 · 4, 6, 2 · 4, 10, 8, 2 · 4, 14, 18, 10, 2 … row sums 6, 12, 24, 48, 96, 192, 384, 768, …
  (−∞, ∞) nonnegative rational roots
• A304236 → θ+1
  1 · 1 · 1, 6 · 1, 12 … row sums 1, 1, 7, 13, 46, 106, 301, 739, …
  (−∞, 0] nonnegative
• A163626: Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).
  1 · 1, -1 · 1, -3, 2 · 1, -7, 12, -6 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  A163626 (0, 1]
• A113214 → x ↦ x−7
  1 · -5, 1 · 28, -11, 1 · -161, 93, -17, 1 … row sums 1, -4, 18, -84, 396, -1872, 8856, …
  (3, 7]
• A082985 → Veronese 2 (j=1)
  3 · 5 · 7, 7 · 9, 30 … row sums 0, 3, 5, 14, 39, 99, 260, 683, …
  (−∞, 0] nonnegative
• A274498 → basis A013610
  1 · 3 · 9, 18 · 27, 54 … row sums 1, 3, 27, 81, 729, 2187, 19683, 59049, …
  (−∞, 0] nonnegative rational roots
• A116414 → D+1
  1 · 5, 1 · 21, 10, 1 · 82, 66, 15, 1 … row sums 1, 6, 32, 164, 819, 4012, 19363, 92352, …
  (−∞, 0] nonnegative