A132812: Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).

OEIS matches ?

• A132812: reversed triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?
• A001791: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A132812: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:  1;
  1:  2,   2;
  2:  3,   9,    3;
  3:  4,  24,   24,     4;
  4:  5,  50,  100,    50,     5;
  5:  6,  90,  300,   300,    90,     6;
  6:  7, 147,  735,  1225,   735,   147,     7;
  7:  8, 224, 1568,  3920,  3920,  1568,   224,    8;
  8:  9, 324, 3024, 10584, 15876, 10584,  3024,  324,   9;
  9: 10, 450, 5400, 25200, 52920, 52920, 25200, 5400, 450, 10;

The row polynomials satisfy (n+2)*P(n,x) = (3*n+3+(3*n+3)*x)*P(n-1,x) - (3*n+(2*n-4)*x+3*n*x^2)*P(n-2,x) + (n-1-(n-1)*x-(n-1)*x^2+(n-1)*x^3)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 3 + 9*x + 3*x^2.

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 3 + 9*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 3 + (3*n + 3)*x)*P[n - 1, x] - (3*n + (2*n - 4)*x + 3*n*x^2)*P[n - 2, x] + (n - 1 - (n - 1)*x - (n - 1)*x^2 + (n - 1)*x^3)*P[n - 3, x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 3 + 9*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n + 3 + (3*n + 3)*x)*v[i - 1] - (3*n + (2*n - 4)*x + 3*n*x^2)*v[i - 2] + (n - 1 - (n - 1)*x - (n - 1)*x^2 + (n - 1)*x^3)*v[i - 3])/(n + 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Triangle begins:
   1:  1;
   2:  2,   2;
   3:  3,   9,    3;
   4:  4,  24,   24,     4;
   5:  5,  50,  100,    50,     5;
   6:  6,  90,  300,   300,    90,     6;
   7:  7, 147,  735,  1225,   735,   147,     7;
   8:  8, 224, 1568,  3920,  3920,  1568,   224,    8;
   9:  9, 324, 3024, 10584, 15876, 10584,  3024,  324,   9;
  10: 10, 450, 5400, 25200, 52920, 52920, 25200, 5400, 450, 10;

The row polynomials satisfy (n+1)*P(n,x) = (3*n+3*n*x)*P(n-1,x) - (3*n-3+(2*n-6)*x+(3*n-3)*x^2)*P(n-2,x) + (n-2-(n-2)*x-(n-2)*x^2+(n-2)*x^3)*P(n-3,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2 + 2*x, P(3,x) = 3 + 9*x + 3*x^2.

P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[3, x_] := 3 + 9*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 3*n*x)*P[n - 1, x] - (3*n - 3 + (2*n - 6)*x + (3*n - 3)*x^2)*P[n - 2, x] + (n - 2 - (n - 2)*x - (n - 2)*x^2 + (n - 2)*x^3)*P[n - 3, x])/(n + 1)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 3 + 9*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = ((3*n + 3*n*x)*v[i - 1] - (3*n - 3 + (2*n - 6)*x + (3*n - 3)*x^2)*v[i - 2] + (n - 2 - (n - 2)*x - (n - 2)*x^2 + (n - 2)*x^3)*v[i - 3])/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Triangle begins:
   1:  1;
   2:  2,   2;
   3:  3,   9,    3;
   4:  4,  24,   24,     4;
   5:  5,  50,  100,    50,     5;
   6:  6,  90,  300,   300,    90,     6;
   7:  7, 147,  735,  1225,   735,   147,     7;
   8:  8, 224, 1568,  3920,  3920,  1568,   224,    8;
   9:  9, 324, 3024, 10584, 15876, 10584,  3024,  324,   9;
  10: 10, 450, 5400, 25200, 52920, 52920, 25200, 5400, 450, 10;

Row n lists the coefficients of P(n,x) in descending powers of x, where (n+1)*P(n,x) = (3*n+3*n*x)*P(n-1,x) - (3*n-3+(2*n-6)*x+(3*n-3)*x^2)*P(n-2,x) + (n-2-(n-2)*x-(n-2)*x^2+(n-2)*x^3)*P(n-3,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2 + 2*x, P(3,x) = 3 + 9*x + 3*x^2.

P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[3, x_] := 3 + 9*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 3*n*x)*P[n - 1, x] - (3*n - 3 + (2*n - 6)*x + (3*n - 3)*x^2)*P[n - 2, x] + (n - 2 - (n - 2)*x - (n - 2)*x^2 + (n - 2)*x^3)*P[n - 3, x])/(n + 1)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 3 + 9*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = ((3*n + 3*n*x)*v[i - 1] - (3*n - 3 + (2*n - 6)*x + (3*n - 3)*x^2)*v[i - 2] + (n - 2 - (n - 2)*x - (n - 2)*x^2 + (n - 2)*x^3)*v[i - 3])/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-041eea8fc0a7af05, e-a32917d44ac940a7, e-fcd77baa2d5e1258
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-041eea8fc0a7af05, e-a32917d44ac940a7, e-f8a41d41829f0ed1, e-fcd77baa2d5e1258
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-041eea8fc0a7af05, e-a32917d44ac940a7, e-f8a41d41829f0ed1, e-fcd77baa2d5e1258
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-f8a41d41829f0ed1
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-f8a41d41829f0ed1
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-a32917d44ac940a7
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-138e1ca38baf9fe5

Proof chains

7 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A007318 e-a32917d44ac940a7

  Geometric basis map (elementary) polytool ?

  • premise (sequence): basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]
    • preserver_image affine, shift 0 e-676bb32d6a1b9c64

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • preserver_image euler_polynomial, shift 1 e-041eea8fc0a7af05

          Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

          • premise (sequence): narayana
            uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check three_term_recurrence e-46a461774e271ec3 root: theorem check

              Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

            • preserver_image reversal, shift 0 e-29ef477df4172e22

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): narayana
                uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check three_term_recurrence e-46a461774e271ec3 (expanded above)
        • preserver_image reversal, shift 0 e-fcd77baa2d5e1258

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ preserver_image euler_polynomial, shift 1 e-041eea8fc0a7af05 (expanded above)
            • theorem_check liu_wang e-f8a41d41829f0ed1 root: theorem check

              Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

        • ↵ theorem_check liu_wang e-f8a41d41829f0ed1 (expanded above)
• definition_equality e-17841f9b7ad2ac5d root: theorem check

  Ore-algebra identity of two definitions polytool ?

• definition_equality e-dee9185888bdea38 root: theorem check

  Ore-algebra identity of two definitions polytool ?

• ↵ preserver_image euler_polynomial, shift 1 e-041eea8fc0a7af05 (expanded above)
• preserver_image affine, shift 0 e-138e1ca38baf9fe5

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • ↵ preserver_image affine, shift 0 e-676bb32d6a1b9c64 (expanded above)
    • theorem_check three_term_recurrence e-72ab2c9b35ff70ef root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

• ↵ preserver_image reversal, shift 0 e-fcd77baa2d5e1258 (expanded above)
• ↵ theorem_check liu_wang e-f8a41d41829f0ed1 (expanded above)

Used in 6 proof steps

• e-cc6de87a980d771e gamma_polynomial (scale 1, shift 0) → gamma_polynomial of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k…
• e-60496e4e2f616741 preserver_image (derivative_polynomial, shift 0) → derivative of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
• e-676bb32d6a1b9c64 preserver_image (affine, shift 0) → basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^…
• e-6a8f8db3c1596aaf preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k…
• e-f4ea45816c9a82eb preserver_image (affine, shift 0) → shift_x_plus_one of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k…
• e-fcd77baa2d5e1258 preserver_image (reversal, shift 0) → Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).

Related sequences ?

Same fingerprint (3)

Equal up to content, sign, reversal and a small index shift.

• 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 Ca…
• A367270 → D
• narayana

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023