definition_equality

All-n equality of two definitions. All operations

163 sequences

• A099040 → basis A007318
  1 · 2, 2 · 6, 10, 4 · 16, 40, 32, 8 … row sums 1, 4, 20, 96, 464, 2240, 10816, 52224, …
  (−∞, −1] nonnegative
• A082985 → x ↦ x+1
  1 · 4, 3 · 11, 15, 5 · 29, 56, 35, 7 … row sums 1, 7, 31, 127, 511, 2047, 8191, 32767, …
  (−∞, −1] nonnegative
• A099093: Riordan array (1, 3+3x).
  1 · 0, 3 · 0, 3, 9 · 0, 0, 18, 27 … row sums 1, 3, 12, 45, 171, 648, 2457, 9315, …
  A099093 (−4/3, 0] nonnegative
• A063007 → x ↦ x+1
  1 · 3, 2 · 13, 18, 6 · 63, 132, 90, 20 … row sums 1, 5, 37, 305, 2641, 23525, 213445, …
  (−2, −1) nonnegative
• A056939: Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.
  A108838 → ⊙ C(n+1,k+1) (shift +1)
  1 · 1, 1 · 1, 4, 1 · 1, 10, 10, 1 … row sums 1, 2, 6, 22, 92, 422, 2074, 10754, …
  A056939 (−∞, 0] nonnegative
• A113953 → basis A007318
  1 · 1, 1 · 3, 4, 1 · 5, 11, 7, 1 … row sums 1, 2, 8, 24, 80, 256, 832, 2688, …
  (−∞, −1] nonnegative
• A033282 → x ↦ x+1
  1 · 3, 2 · 11, 15, 5 · 45, 93, 63, 14 … row sums 1, 5, 31, 215, 1597, 12425, 99955, …
  (−2, −1) nonnegative
• A100862: Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n…
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … row sums 1, 2, 5, 14, 43, 142, 499, 1850, …
  A100862 (−∞, 0) nonnegative
• A154986: Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.
  1 · 1, 1 · 1, 4, 1 · 1, 11, 11, 1 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A154986 (−∞, 0) nonnegative
• A008277 → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 1, 28, 54, 16 … row sums 1, 5, 22, 99, 471, 2386, 12867, 73681, …
  (−∞, 0] nonnegative
• A081579 → basis A007318
  1 · 2, 1 · 8, 8, 1 · 24, 36, 14, 1 … row sums 1, 3, 17, 75, 361, 1683, 7937, 37275, …
  (−∞, −1] nonnegative
• 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
• A128099 → x ↦ x+1
  1 · 1 · 3, 2 · 5, 4 … row sums 1, 1, 5, 9, 29, 65, 181, 441, …
  (−∞, −1] nonnegative
• A145596: basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x -> x + 1
  1 · 2, 2 · 3, 8, 3 · 4, 20, 20, 4 … row sums 1, 4, 14, 48, 165, 572, 2002, 7072, …
  A145596 (−∞, 0) nonnegative
• A104684 → basis A007318
  1 · 3, 1 · 13, 8, 1 · 63, 57, 15, 1 … row sums 1, 4, 22, 136, 886, 5944, 40636, …
  (−∞, −1] nonnegative
• A100861 → reverse → x ↦ x+1
  1 · 1, 1 · 2, 3, 1 · 4, 9, 6, 1 … row sums 1, 2, 6, 20, 76, 312, 1384, 6512, …
  (−∞, −1] nonnegative
• A116412 → basis A007318
  1 · 4, 1 · 13, 8, 1 · 43, 42, 12, 1 … row sums 1, 5, 22, 98, 436, 1940, 8632, 38408, …
  (−∞, −1] nonnegative
• A141611: Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).
  A007318 → −D²+(−1+n)D+1+n
  1 · 2, 2 · 3, 8, 3 · 4, 18, 18, 4 … row sums 1, 4, 14, 44, 128, 352, 928, 2368, …
  A141611 (−∞, 0] nonnegative
• A297191: Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.
  basis-map coefficients of A297191, shifted → basis A008288
  1 · 1, 3, 1 · 1, 7, 13, 7, 1 … row sums 1, 5, 29, 169, 985, 5741, 33461, …
  A297191 [−3−2√2, −3+2√2] nonnegative
• A055151: Triangular array of Motzkin polynomial coefficients.
  narayana → γ (shift +1)
  1 · 1 · 1, 1 · 1, 3 … row sums 1, 1, 2, 4, 9, 21, 51, 127, …
  A055151 (−∞, 0] nonnegative
• A274498 → x ↦ x+1
  1 · 3 · 9, 6 · 27, 18 … row sums 1, 3, 15, 45, 225, 675, 3375, 10125, …
  (−∞, −1] 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
• A059473: Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...
  1 · 2, 2 · 4, 10, 4 · 8, 32, 32, 8 … row sums 1, 4, 18, 80, 356, 1584, 7048, 31360, …
  A059473 (−∞, 0) nonnegative
• A132440: Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the lis…
  A023531 → D
  1 · 0, 2 · 0, 0, 3 · 0, 0, 0, 4 … row sums 0, 1, 2, 3, 4, 5, 6, 7, …
  A132440 (−∞, 0] nonnegative rational roots
• A143683: Pascal-(1,8,1) array.
  1 · 1, 1 · 1, 10, 1 · 1, 19, 19, 1 … row sums 1, 2, 12, 40, 176, 672, 2752, 10880, …
  A143683 (−∞, 0) nonnegative
• A081582: Pascal-(1,7,1) array.
  1 · 1, 1 · 1, 9, 1 · 1, 17, 17, 1 … row sums 1, 2, 11, 36, 149, 550, 2143, 8136, …
  A081582 (−∞, 0) nonnegative
• A099093 → basis A007318
  1 · 3, 3 · 12, 21, 9 · 45, 117, 99, 27 … row sums 1, 6, 42, 288, 1980, 13608, 93528, …
  (−∞, −1] nonnegative
• A249252 → basis A007318
  1 · 5, 4 · 17, 24, 8 · 61, 124, 80, 16 … row sums 1, 9, 49, 281, 1601, 9129, 52049, …
  (−∞, −1] nonnegative
• A081581: Pascal-(1,6,1) array.
  1 · 1, 1 · 1, 8, 1 · 1, 15, 15, 1 … row sums 1, 2, 10, 32, 124, 440, 1624, 5888, …
  A081581 (−∞, 0) nonnegative
• A249250 → x ↦ x+1
  1 · 4, 3 · 10, 12, 3 · 28, 48, 24, 3 … row sums 1, 7, 25, 103, 409, 1639, 6553, 26215, …
  (−∞, −1] nonnegative
• A030528: Triangle read by rows: T(n,k) = binomial(k,n-k).
  1 · 1, 1 · 0, 2, 1 · 0, 1, 3, 1 … row sums 1, 2, 3, 5, 8, 13, 21, 34, …
  A030528 (−4, 0] nonnegative
• A091044: One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
  1 · 2, 2 · 3, 10, 3 · 4, 28, 28, 4 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A091044 (−∞, 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
• 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
• 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
• A143685 → basis A007318
  1 · 2, 1 · 13, 13, 1 · 44, 66, 24, 1 … row sums 1, 3, 27, 135, 891, 5103, 31347, …
  (−∞, −1] nonnegative
• 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
• A216973: Exponential Riordan array [x*exp(x),x].
  A007318 → D
  1 · 2, 2 · 3, 6, 3 · 4, 12, 12, 4 … row sums 0, 1, 4, 12, 32, 80, 192, 448, …
  A216973 (−∞, −1] 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
• A116414 → basis A007318
  1 · 5, 1 · 22, 10, 1 · 95, 69, 15, 1 … row sums 1, 6, 33, 180, 981, 5346, 29133, …
  (−9, −1) 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
• A249251 → basis A007318
  1 · 4, 3 · 13, 18, 6 · 43, 87, 57, 12 … row sums 1, 7, 37, 199, 1069, 5743, 30853, …
  (−∞, −1] nonnegative
• A135065: A127733 * A007318 as infinite lower triangular matrices.
  A007318 → ×(1+2n+n²)
  1 · 4, 4 · 9, 18, 9 · 16, 48, 48, 16 … row sums 1, 8, 36, 128, 400, 1152, 3136, 8192, …
  A135065 (−∞, −1] 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
• A108838 → x ↦ x+1
  2 · 5, 2 · 14, 12, 2 · 42, 56, 21, 2 … row sums 2, 7, 28, 121, 550, 2591, 12536, 61921, …
  (−∞, −1] nonnegative
• A081577 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 16, 24, 10, 1 … row sums 1, 3, 13, 51, 205, 819, 3277, 13107, …
  (−∞, −1] nonnegative
• A113413 → basis A007318
  1 · 3, 1 · 7, 6, 1 · 17, 23, 9, 1 … row sums 1, 4, 14, 50, 178, 634, 2258, 8042, …
  (−∞, −1] nonnegative
• A081581 → basis A007318
  1 · 2, 1 · 10, 10, 1 · 32, 48, 18, 1 … row sums 1, 3, 21, 99, 549, 2835, 15093, 79299, …
  (−∞, −1] 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