A085881 → D+1

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Submission text ?

Triangle begins:
  0:         1;
  1:         2,          1;
  2:         9,         12,          3;
  3:        60,        135,         90,          15;
  4:       525,       1680,       1890,         840,         105;
  5:      5670,      23625,      37800,       28350,        9450,         945;
  6:     72765,     374220,     779625,      831600,      467775,      124740,       10395;
  7:   1081080,    6621615,   17027010,    23648625,    18918900,     8513505,     1891890,     135135;
  8:  18243225,  129729600,  397296900,   681080400,   709458750,   454053600,   170270100,   32432400,   2027025;
  9: 344594250, 2791213425, 9924314400, 20262141900, 26051325300, 21709437750, 11578366800, 3721617900, 620269650, 34459425;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + n - 1 + (2*n^2 + 3*n - 2)*x + (2*n - 1)*x^2)*v[i - 1])/(n + x)); 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)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
    • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
    • Parent-row provers of polytool polytool ? step › direct check
    • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A178821 → basis A013609 → θ+1
• A085880 → θ+1
• A135838 → D+1
• basis-map coefficients of A026729 → basis A007318 → θ+1
• basis-map coefficients of A127647 → θ+1
• A085881 → basis A013609 → θ+1
• A196347 → θ+1
• A178820 → θ+1
• basis-map coefficients of A113214, shifted → basis A013609 → D+1
• A003506 → D → θ+1
• A038221 → basis A013609 → D+1
• A027465 → basis A013611 → θ+1
• basis-map coefficients of A113214, shifted → D → θ+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• basis-map coefficients of A026729 → basis A007318 → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A085880 → basis A013609 → D+1
• A038338 → basis A013609 → θ+1
• A038338 → D+1
• A038338 → basis A013609 → D+1
• A356546 → θ+1
• A120910 → basis A013610 → D+1
• A038208 → D → D+1
• A085881 → basis A013609 → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• basis-map coefficients of A113214, shifted → D → D+1
• A038208 → D+1
• basis-map coefficients of A026729, shifted → basis A007318 → D+1
• A120910 → basis A013610 → θ+1
• A178820 → D+1
• A038208 → D → θ+1
• A038325 → D+1
• A038330 → basis A013611 → D+1
• A135838 → θ+1
• A038208 → θ+1
• A178820 → basis A013609 → θ+1
• A038325 → basis A013609 → θ+1
• A135065 → θ+1
• A124860 → D+1
• A028326 → D+1
• A085880 → D → θ+1
• A028326 → basis A013609 → D+1
• A038325 → θ+1
• A156992 → θ+1
• A196347 → basis A013609 → D+1
• A093375 → D
• basis-map coefficients of A026729, shifted → basis A007318 → θ+1
• A131431 → D+1 → basis A007318
• A131431 → x ↦ x+1 → θ+1
• A028326 → D → θ+1
• A131431 → x ↦ x+1 → D+1
• A028326 → θ+1
• A038221 → θ+1
• A038330 → basis A013611 → θ+1
• basis-map coefficients of A113214, shifted → θ+1
• A376827 → θ+1
• A178820 → basis A013609 → D+1
• A376827 → D+1
• A178822 → basis A013609 → D+1
• A085880 → basis A013609 → θ+1
• A178821 → D+1
• A178822 → D+1
• A178821 → θ+1
• A356546 → D+1
• A356546 → basis A013609 → D+1
• A038338 → θ+1
• A038325 → basis A013609 → D+1
• A156992 → basis A013609 → D+1
• basis-map coefficients of A112227 → x ↦ x+1 → θ+1
• A178822 → θ+1
• A038221 → D+1
• A356546 → basis A013609 → θ+1
• A196347 → basis A013609 → θ+1
• A216973 → θ+1
• A134346 → D+1
• basis-map coefficients of A127647 → D+1
• A027465 → basis A013611 → D+1
• A156992 → basis A013609 → θ+1
• A178821 → basis A013609 → D+1
• A085880 → D → D+1
• A028326 → D → D+1
• basis-map coefficients of A113214, shifted → basis A013609 → θ+1
• A038221 → basis A013609 → θ+1
• A028326 → basis A013609 → θ+1
• A094305 → θ+1
• A094305 → D+1
• A196347 → D+1
• A124860 → θ+1
• A085880 → D+1
• A093375 → D → D
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A344565 → θ+1
• A156992 → D+1
• basis-map coefficients of A113214, shifted → D+1
• A344565 → D+1
• A134346 → θ+1
• A135065 → D+1
• A178822 → basis A013609 → θ+1
• A085881 → θ+1
• basis-map coefficients of A112227 → x ↦ x+1 → D+1

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
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.