A356546 → (θ+1)²

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Submission text ?

Triangle begins:
  0:     1;
  1:     2,       8;
  2:     6,      48,       54;
  3:    20,     240,      540,      320;
  4:    70,    1120,     3780,     4480,      1750;
  5:   252,    5040,    22680,    40320,     31500,      9072;
  6:   924,   22176,   124740,   295680,    346500,    199584,     45276;
  7:  3432,   96096,   648648,  1921920,   3003000,   2594592,   1177176,    219648;
  8: 12870,  411840,  3243240, 11531520,  22522500,  25945920,  17657640,   6589440,  1042470;
  9: 48620, 1750320, 15752880, 65345280, 153153000, 220540320, 200119920, 112020480, 35443980, 4862000;

The row polynomials satisfy (n+(3*n^2-n)*x+n^3*x^2)*P(n,x) = (4*n-2+(12*n^2+6*n-6)*x+(4*n^3+18*n^2+2*n-6)*x^2+(4*n^3+6*n^2-2)*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[((4*n - 2 + (12*n^2 + 6*n - 6)*x + (4*n^3 + 18*n^2 + 2*n - 6)*x^2 + (4*n^3 + 6*n^2 - 2)*x^3)*P[n - 1, x])/(n + (3*n^2 - n)*x + n^3*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] = ((4*n - 2 + (12*n^2 + 6*n - 6)*x + (4*n^3 + 18*n^2 + 2*n - 6)*x^2 + (4*n^3 + 6*n^2 - 2)*x^3)*v[i - 1])/(n + (3*n^2 - n)*x + n^3*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 read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
    • 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

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