A085881 → (θ+1)²

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Submission text ?

Triangle begins:
  0:        1;
  1:        1,          4;
  2:        3,         24,          27;
  3:       15,        180,         405,         240;
  4:      105,       1680,        5670,        6720,         2625;
  5:      945,      18900,       85050,      151200,       118125,        34020;
  6:    10395,     249480,     1403325,     3326400,      3898125,      2245320,       509355;
  7:   135135,    3783780,    25540515,    75675600,    118243125,    102162060,     46351305,     8648640;
  8:  2027025,   64864800,   510810300,  1816214400,   3547293750,   4086482400,   2781078300,  1037836800,   164189025;
  9: 34459425, 1240539300, 11164853700, 46313467200, 108547188750, 156307951800, 141834993300, 79394515200, 25120920825, 3445942500;

The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = (2*n-1+(6*n^2+3*n-3)*x+(2*n^3+9*n^2+n-3)*x^2+(2*n^3+3*n^2-1)*x^3)*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 - 1 + (6*n^2 + 3*n - 3)*x + (2*n^3 + 9*n^2 + n - 3)*x^2 + (2*n^3 + 3*n^2 - 1)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*x + n^2*x^2)]];
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 - 1 + (6*n^2 + 3*n - 3)*x + (2*n^3 + 9*n^2 + n - 3)*x^2 + (2*n^3 + 3*n^2 - 1)*x^3)*v[i - 1])/(1 + (3*n - 1)*x + n^2*x^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)

Properties

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

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 ?

• A120910 → basis A013610 → (θ+1)²
• basis-map coefficients of A026729 → basis A007318 → (θ+1)²
• A135065 → (θ+1)²
• A344565 → (θ+1)²
• A178821 → basis A013609 → (θ+1)²
• basis-map coefficients of A026729, shifted → basis A007318 → (θ+1)²
• A178822 → (θ+1)²
• A085880 → D → (θ+1)²
• A038330 → basis A013611 → (θ+1)²
• A356546 → (θ+1)²
• A003506 → (θ+1)²
• basis-map coefficients of A113214, shifted → basis A013609 → (θ+1)²
• basis-map coefficients of A113214, shifted → (θ+1)²
• A085881 → basis A013609 → (θ+1)²
• basis-map coefficients of A127647 → (θ+1)²
• A027465 → basis A013611 → (θ+1)²
• A038221 → basis A013609 → (θ+1)²
• A131431 → x ↦ x+1 → (θ+1)²
• A125092 → reverse → D
• A028326 → D → (θ+1)²
• A125092 → reverse
• A038338 → basis A013609 → (θ+1)²
• A038325 → basis A013609 → (θ+1)²
• A038221 → (θ+1)²
• A038208 → (θ+1)²
• A356546 → basis A013609 → (θ+1)²
• A124860 → (θ+1)²
• A038208 → D → (θ+1)²
• A376827 → (θ+1)²
• A156992 → basis A013609 → (θ+1)²
• basis-map coefficients of A113214, shifted → D → (θ+1)²
• A038325 → (θ+1)²
• A038338 → (θ+1)²
• A178820 → basis A013609 → (θ+1)²
• A178822 → basis A013609 → (θ+1)²
• A196347 → basis A013609 → (θ+1)²
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
• A178821 → (θ+1)²
• A003506 → D → (θ+1)²
• A135838 → (θ+1)²
• A085880 → basis A013609 → (θ+1)²
• A156992 → (θ+1)²
• basis-map coefficients of A112227 → x ↦ x+1 → (θ+1)²
• A028326 → basis A013609 → (θ+1)²
• A178820 → (θ+1)²
• A028326 → (θ+1)²
• A196347 → (θ+1)²
• A085880 → (θ+1)²
• A134346 → (θ+1)²

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.