definition_equality

All-n equality of two definitions. All operations

163 sequences

• A091044 → basis A007318
  1 · 4, 2 · 16, 16, 3 · 64, 96, 40, 4 … row sums 1, 6, 35, 204, 1189, 6930, 40391, …
  (−∞, −1] nonnegative
• A152198: Triangle read by rows, A007318 rows repeated.
  1 · 1 · 1, 1 · 1, 1 … row sums 1, 1, 2, 2, 4, 4, 8, 8, …
  A152198 A152815 (−∞, 0] nonnegative rational roots
• A145597 → basis A007318
  1 · 6, 3 · 27, 27, 6 · 110, 165, 75, 10 … row sums 1, 9, 60, 360, 2061, 11529, 63728, …
  (−∞, −1] nonnegative
• A059473 → x ↦ x+1
  1 · 4, 2 · 18, 18, 4 · 80, 120, 56, 8 … row sums 1, 6, 40, 264, 1744, 11520, 76096, …
  (−∞, −1] 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
• basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). 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
• A123519 → x ↦ x+1
  11, 14, 4 · 41, 76, 44, 8 · 153, 372, 324, 120, 16 … row sums 29, 169, 985, 5741, 33461, 195025, …
  (−3, −1) nonnegative
• A304236 → basis A007318
  1 · 1 · 4, 3 · 7, 6 … row sums 1, 1, 7, 13, 55, 133, 463, 1261, …
  (−∞, −1] nonnegative
• A188440 → basis A007318
  1 · 1 · 3, 2 · 3, 2 … row sums 1, 1, 5, 5, 25, 25, 125, 125, …
  (−∞, −1] nonnegative rational roots
• A081580 → basis A007318
  1 · 2, 1 · 9, 9, 1 · 28, 42, 16, 1 … row sums 1, 3, 19, 87, 451, 2223, 11179, 55767, …
  (−∞, −1] nonnegative
• A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
  A010054 → reverse
  1 · 0, 3 · 0, 0, 6 · 0, 0, 0, 10 … row sums 1, 3, 6, 10, 15, 21, 28, 36, …
  A127773 [−1, 0] nonnegative rational roots
• A030528 → basis A007318
  1 · 2, 1 · 3, 4, 1 · 5, 10, 6, 1 … row sums 1, 3, 8, 22, 60, 164, 448, 1224, …
  (−∞, −1] nonnegative
• A010054 → reverse → reverse
  1 · 3 · 6 · 10 … row sums 1, 3, 6, 10, 15, 21, 28, 36, …
  (−∞, 0] nonnegative rational roots
• A169803: Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).
  1 · 1, 1 · 1, 2 · 1, 3, 1 … row sums 1, 2, 3, 5, 8, 13, 21, 34, …
  A169803 (−∞, 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
• A008459 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 20, 30, 12, 1 … row sums 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  (−∞, −1] nonnegative
• A099097: Riordan array (1, 3+x).
  1 · 0, 3 · 0, 1, 9 · 0, 0, 6, 27 … row sums 1, 3, 10, 33, 109, 360, 1189, 3927, …
  A099097 (−4/9, 0] nonnegative
• basis-map coefficients of A scaled Hermite triangle.
  0, 0, 1 · 0, 0, 0, 1 · 0, 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0] nonnegative rational roots
• A052553 → x ↦ x+1
  1 · 1 · 2, 1 · 3, 2 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  (−∞, −1] nonnegative
• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
  A010054 → D∘x
  1 · 3 · 5 · 7 … row sums 1, 3, 5, 7, 9, 11, 13, 15, …
  A198954 (−∞, −1] [−1, 0] nonnegative rational roots
• A033282: Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
  1 · 1, 2 · 1, 5, 5 · 1, 9, 21, 14 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A033282 (−1, 0) nonnegative
• A054142: Triangular array binomial(2*n-k, k), k=0..n, n >= 0.
  1 · 1, 1 · 1, 3, 1 · 1, 5, 6, 1 … row sums 1, 2, 5, 13, 34, 89, 233, 610, …
  A054142 (−∞, 0) nonnegative
• A111125 → x ↦ x+1
  1 · 4, 1 · 11, 7, 1 · 29, 31, 10, 1 … row sums 1, 5, 19, 71, 265, 989, 3691, 13775, …
  (−5, −1) nonnegative
• A152842 → x ↦ x+1
  1 · 4, 3 · 8, 10, 3 · 32, 64, 42, 9 … row sums 1, 7, 21, 147, 441, 3087, 9261, 64827, …
  (−∞, −1] nonnegative rational roots
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
  1 · 1, 3 · 1, 6, 9 · 1, 9, 27, 27 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A013610 (−∞, 0) [−1, 0] nonnegative rational roots
• A081580: Pascal-(1,5,1) array.
  1 · 1, 1 · 1, 7, 1 · 1, 13, 13, 1 … row sums 1, 2, 9, 28, 101, 342, 1189, 4088, …
  A081580 (−∞, 0) nonnegative
• A114656 → basis A007318
  1 · 3, 1 · 11, 8, 1 · 45, 51, 15, 1 … row sums 1, 4, 20, 112, 672, 4224, 27456, …
  (−∞, −1] nonnegative
• A100862 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … row sums 1, 3, 11, 45, 201, 963, 4899, 26253, …
  (−∞, −1] nonnegative
• A110441 → basis A007318
  1 · 4, 1 · 14, 8, 1 · 48, 44, 12, 1 … row sums 1, 5, 23, 105, 479, 2185, 9967, 45465, …
  (−7, −1) nonnegative
• narayana → basis A007318
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  (−∞, −1] nonnegative
• A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
  1 · 0, 1 · 0, 0, 1 · 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A023531 [−1, 0] [0, ∞) nonnegative rational roots
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
  1 · 1, 2 · 1, 4, 4 · 1, 6, 12, 8 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A013609 (−∞, 0) [−1, 0] nonnegative rational roots
• A064861 → x ↦ x+1
  6, 7, 2 · 18, 33, 20, 4 · 36, 84, 73, 28, 4 … row sums 15, 75, 225, 1125, 3375, 16875, 50625, …
  (−∞, −1] nonnegative rational roots
• A172431 → basis A007318
  1 · 3, 2 · 8, 10, 3 · 21, 38, 22, 4 … row sums 1, 5, 21, 85, 341, 1365, 5461, 21845, …
  (−∞, −1] nonnegative
• A011973: Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.
  1 · 1 · 1, 1 · 1, 2 … row sums 1, 1, 2, 3, 5, 8, 13, 21, …
  A011973 (−∞, 0] nonnegative
• A104684: Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.
  1 · 2, 1 · 6, 6, 1 · 20, 30, 12, 1 … row sums 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  A104684 (−∞, 0) nonnegative
• A143685: Pascal-(1,9,1) array.
  1 · 1, 1 · 1, 11, 1 · 1, 21, 21, 1 … row sums 1, 2, 13, 44, 205, 806, 3457, 14168, …
  A143685 (−∞, 0) nonnegative
• A128100: Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).
  A011973 → basis A007318
  1 · 1 · 2, 1 · 3, 2 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  A128100 (−∞, −1] nonnegative
• A094305: Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).
  A007318 → ×(1+(3/2)n+(1/2)n²)
  1 · 3, 3 · 6, 12, 6 · 10, 30, 30, 10 … row sums 1, 6, 24, 80, 240, 672, 1792, 4608, …
  A094305 (−∞, −1] nonnegative rational roots
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.
  A007318 → θ+1
  1 · 1, 2 · 1, 4, 3 · 1, 6, 9, 4 … row sums 1, 3, 8, 20, 48, 112, 256, 576, …
  A103406 (−∞, 0] nonnegative rational roots
• A008288: Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
  1 · 1, 1 · 1, 3, 1 · 1, 5, 5, 1 … row sums 1, 2, 5, 12, 29, 70, 169, 408, …
  A008288 [−3−2√2, −3+2√2] nonnegative
• A100861 → basis A007318
  1 · 1 · 2, 1 · 4, 3 … row sums 1, 1, 3, 7, 25, 81, 331, 1303, …
  (−∞, −1] nonnegative
• A126216 → basis A007318
  1 · 3, 1 · 11, 7, 1 · 45, 42, 12, 1 … row sums 1, 4, 19, 100, 562, 3304, 20071, …
  (−∞, −1] nonnegative
• A088996: Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.
  1 · 0, 1 · 0, 1, 2 · 0, 2, 7, 6 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  A088996 [−1, 0] nonnegative rational roots
• A008288 → basis A007318
  1 · 2, 1 · 5, 5, 1 · 12, 18, 8, 1 … row sums 1, 3, 11, 39, 139, 495, 1763, 6279, …
  (−∞, −1] nonnegative
• A103327 → basis A007318
  1 · 4, 1 · 16, 12, 1 · 64, 80, 24, 1 … row sums 1, 5, 29, 169, 985, 5741, 33461, …
  (−∞, −1] nonnegative
• A059344 → basis A007318
  1 · 1 · 3, 2 · 7, 6 … row sums 1, 1, 5, 13, 73, 281, 1741, 8485, …
  (−∞, −1] nonnegative
• A391403: Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).
  A011973 → ⊙ C(n,k)
  1 · 1 · 1, 2 · 1, 6 … row sums 1, 1, 3, 7, 19, 51, 141, 393, …
  A391403 (−∞, 0] nonnegative
• A152198 → x ↦ x+1
  1 · 1 · 2, 1 · 2, 1 … row sums 1, 1, 3, 3, 9, 9, 27, 27, …
  (−∞, −1] nonnegative rational roots