definition_equality

All-n equality of two definitions. All operations

163 sequences

• A007318 → D → D+1
  1 · 4, 2 · 9, 12, 3 · 16, 36, 24, 4 … row sums 0, 1, 6, 24, 80, 240, 672, 1792, …
  (−∞, 0] nonnegative rational roots
• A100861: Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
  1 · 1 · 1, 1 · 1, 3 … row sums 1, 1, 2, 4, 10, 26, 76, 232, …
  A100861 A144299 (−∞, 0] nonnegative
• A132812: Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
  1 · 2, 2 · 3, 9, 3 · 4, 24, 24, 4 … row sums 1, 4, 15, 56, 210, 792, 3003, 11440, …
  A132812 (−∞, 0) nonnegative
• A081582 → x ↦ x+1
  1 · 2, 1 · 11, 11, 1 · 36, 54, 20, 1 … row sums 1, 3, 23, 111, 655, 3519, 19727, …
  (−∞, −1] nonnegative
• A237765: Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
  A007318 → ×(−(1/2)n+(1/2)n²)
  1, 2, 1 · 3, 9, 9, 3 · 6, 24, 36, 24, 6 … row sums 0, 0, 4, 24, 96, 320, 960, 2688, …
  A237765 (−∞, 0] nonnegative rational roots
• A059473 → γ
  1 · 2 · 4, 2 · 8, 8 … row sums 1, 2, 6, 16, 44, 120, 328, 896, …
  (−∞, 0] nonnegative
• A099097 → basis A007318
  1 · 3, 3 · 10, 19, 9 · 33, 93, 87, 27 … row sums 1, 6, 38, 240, 1516, 9576, 60488, …
  (−∞, −1] nonnegative
• A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).
  1 · 0, 2 · 0, 0, 4 · 0, 0, 0, 8 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A130123 [−1, 0] nonnegative rational roots
• A169803 → x ↦ x+1
  1 · 2, 1 · 3, 2 · 5, 5, 1 … row sums 1, 3, 5, 11, 21, 43, 85, 171, …
  (−∞, −1] nonnegative
• A111910: Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).
  A091044 → ⊙ C(n+2,k+1)
  1 · 1, 1 · 1, 5, 1 · 1, 14, 14, 1 … row sums 1, 2, 7, 30, 146, 772, 4331, 25398, …
  A111910 (−∞, 0] nonnegative
• A066325: Hermite he
  1 · 0, 1 · -1, 0, 1 · 0, -3, 0, 1 … row sums 1, 1, 0, -2, -2, 6, 16, -20, …
  A066325 (−∞, ∞)
• A008459: Square the entries of Pascal's triangle.
  1 · 1, 1 · 1, 4, 1 · 1, 9, 9, 1 … row sums 1, 2, 6, 20, 70, 252, 924, 3432, …
  A008459 (−∞, 0) nonnegative
• A103633 → basis A007318
  1 · 1, 1 · 2, 3, 1 · 2, 5, 4, 1 … row sums 1, 2, 6, 12, 36, 72, 216, 432, …
  (−∞, −1] nonnegative rational roots
• A143683 → x ↦ x+1
  1 · 2, 1 · 12, 12, 1 · 40, 60, 22, 1 … row sums 1, 3, 25, 123, 769, 4275, 25129, …
  (−∞, −1] nonnegative
• A114687 → basis A007318
  1 · 3, 2 · 11, 14, 4 · 45, 84, 48, 8 … row sums 1, 5, 29, 185, 1257, 8925, 65445, …
  (−∞, −1] nonnegative
• A121574 → basis A007318
  1 · 3, 1 · 10, 7, 1 · 33, 35, 11, 1 … row sums 1, 4, 18, 80, 356, 1584, 7048, 31360, …
  (−∞, −1] nonnegative
• A113953: A Jacobsthal triangle.
  1 · 0, 1 · 0, 2, 1 · 0, 0, 4, 1 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  A113953 (−8, 0] nonnegative
• A145597: Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.
  1 · 3, 3 · 6, 15, 6 · 10, 45, 45, 10 … row sums 1, 6, 27, 110, 429, 1638, 6188, 23256, …
  A145597 (−∞, 0) nonnegative
• A081578 → x ↦ x+1
  1 · 2, 1 · 7, 7, 1 · 20, 30, 12, 1 … row sums 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  (−∞, −1] nonnegative
• A086645 → x ↦ x+1
  1 · 2, 1 · 8, 8, 1 · 32, 48, 18, 1 … row sums 1, 3, 17, 99, 577, 3363, 19601, 114243, …
  (−∞, −1] nonnegative
• A207607 → basis A007318
  1 · 3, 2 · 8, 9, 2 · 21, 33, 15, 2 … row sums 1, 5, 19, 71, 265, 989, 3691, 13775, …
  (−5, −1) nonnegative
• A113214: Riordan array (1+2x,x(1+x)).
  1 · 2, 1 · 0, 3, 1 · 0, 2, 4, 1 … row sums 1, 3, 4, 7, 11, 18, 29, 47, …
  A113214 (−4, 0] nonnegative
• A367270 → x ↦ x+1
  1 · 1 · 3, 2 · 10, 12, 3 … row sums 1, 1, 5, 25, 129, 681, 3653, 19825, …
  (−∞, −1] nonnegative
• A207824: Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).
  1 · 5, 1 · 24, 10, 1 · 115, 73, 15, 1 … row sums 1, 6, 35, 204, 1189, 6930, 40391, …
  A207824 (−7, −3) nonnegative
• A001263: Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … row sums 1, 2, 5, 14, 42, 132, 429, 1430, …
  A001263 (−∞, 0) nonnegative
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
  1 · 3, 5 · 9, 30, 25 · 27, 135, 225, 125 … row sums 1, 8, 64, 512, 4096, 32768, 262144, …
  A013622 (−∞, 0) [−1, 0] nonnegative rational roots
• A063967 → basis A007318
  1 · 2, 1 · 6, 5, 1 · 16, 20, 8, 1 … row sums 1, 3, 12, 45, 171, 648, 2457, 9315, …
  (−∞, −1] nonnegative
• A063967: Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
  1 · 1, 1 · 2, 3, 1 · 3, 7, 5, 1 … row sums 1, 2, 6, 16, 44, 120, 328, 896, …
  A063967 (−5, −1] nonnegative
• A052553: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.
  1 · 1 · 1, 1 · 1, 2 … row sums 1, 1, 2, 3, 5, 8, 13, 21, …
  A052553 (−∞, 0] nonnegative
• A046716: Coefficients of a special case of Poisson-Charlier polynomials.
  1 · 1, 1 · 1, 3, 1 · 1, 6, 8, 1 … row sums 1, 2, 5, 16, 65, 326, 1957, 13700, …
  A046716 (−∞, 0) nonnegative
• A232535 → x ↦ x+1
  1 · 3, 2 · 12, 14, 3 · 48, 80, 37, 4 … row sums 1, 5, 29, 169, 985, 5741, 33461, …
  (−∞, −1] nonnegative
• A131431: 3n + 1 preceded by n zeros.
  A010054 → reverse
  1 · 0, 4 · 0, 0, 7 · 0, 0, 0, 10 … row sums 1, 4, 7, 10, 13, 16, 19, 22, …
  A131431 [−1, 0] nonnegative rational roots
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0
  1 · 0, 0, 1 · 0, 0, 0, 0, 1 · 0, 0, 0, 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0] [0, ∞) nonnegative rational roots
• basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
  1 · 0, -1 · 0, 0, 2 · 0, 0, 0, -6 … row sums 1, -1, 2, -6, 24, -120, 720, -5040, …
  [−1, 0] rational roots
• A229032: Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).
  A011973 → x ↦ x−1/4
  1 · 2 · 3, 4 · 4, 16 … row sums 1, 2, 7, 20, 61, 182, 547, 1640, …
  A229032 (−∞, 0] nonnegative
• A085478 → x ↦ x+1
  1 · 1 · 2, 1 · 5, 5, 1 … row sums 1, 1, 3, 11, 41, 153, 571, 2131, …
  (−5, −1) nonnegative
• A274496 → x ↦ x+1
  1 · 2 · 4, 2 · 8, 4 … row sums 1, 2, 6, 12, 36, 72, 216, 432, …
  (−∞, −1] nonnegative rational roots
• A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
  A023531 → (θ+1)²
  1 · 0, 4 · 0, 0, 9 · 0, 0, 0, 16 … row sums 1, 4, 9, 16, 25, 36, 49, 64, …
  A127733 [−1, 0] nonnegative rational roots
• A116412: Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
  1 · 3, 1 · 6, 6, 1 · 12, 21, 9, 1 … row sums 1, 4, 13, 43, 142, 469, 1549, 5116, …
  A116412 (−∞, 0) nonnegative
• A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
  A010054 → reverse
  1 · 0, 2 · 0, 0, 3 · 0, 0, 0, 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  A127648 [−1, 0] nonnegative rational roots
• A133087: A133080 * A007318.
  1 · 2, 1 · 1, 2, 1 · 2, 5, 4, 1 … row sums 1, 3, 4, 12, 16, 48, 64, 192, …
  A133087 (−∞, −1] nonnegative rational roots
• A081578: Pascal-(1,3,1) array.
  1 · 1, 1 · 1, 5, 1 · 1, 9, 9, 1 … row sums 1, 2, 7, 20, 61, 182, 547, 1640, …
  A081578 (−∞, 0) nonnegative
• A019538: Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
  1 · 1, 2 · 1, 6, 6 · 1, 14, 36, 24 … row sums 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  A019538 (−1, 0) nonnegative
• A007318 → D → γ
  1 · 2 · 3 · 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 0] nonnegative rational roots
• A207606 → basis A007318
  13, 11, 2 · 34, 44, 17, 2 · 89, 155, 93, 23, 2 … row sums 26, 97, 362, 1351, 5042, 18817, 70226, …
  (−5, −1) nonnegative
• delannoy
  1 · 1, 1 · 1, 3, 1 · 1, 5, 5, 1 … row sums 1, 2, 5, 12, 29, 70, 169, 408, …
  [−3−2√2, −3+2√2] nonnegative
• A304236: Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.
  1 · 1 · 1, 3 · 1, 6 … row sums 1, 1, 4, 7, 19, 40, 97, 217, …
  A304236 (−∞, 0] nonnegative
• A128099: Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).
  1 · 1 · 1, 2 · 1, 4 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  A128099 (−∞, 0] nonnegative
• A133087 → x ↦ x+1
  1 · 3, 1 · 4, 4, 1 · 12, 16, 7, 1 … row sums 1, 4, 9, 36, 81, 324, 729, 2916, …
  (−∞, −1] nonnegative rational roots
• A249251: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 3 · 1, 6, 6 · 1, 9, 21, 12 … row sums 1, 4, 13, 43, 142, 469, 1549, 5116, …
  A249251 (−∞, 0) nonnegative