A356546 → x ↦ x+1 → D+1

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Submission text ?

Triangle begins:
  0:         1;
  1:         6,         2;
  2:        48,        36,          6;
  3:       400,       480,        180,         20;
  4:      3360,      5600,       3360,        840,        70;
  5:     28224,     60480,      50400,      20160,      3780,       252;
  6:    236544,    620928,     665280,     369600,    110880,     16632,      924;
  7:   1976832,   6150144,    8072064,    5765760,   2402400,    576576,    72072,     3432;
  8:  16473600,  59304960,   92252160,   80720640,  43243200,  14414400,  2882880,   308880,   12870;
  9: 136913920, 560102400, 1008184320, 1045524480, 686125440, 294053760, 81681600, 14002560, 1312740, 48620;

The row polynomials satisfy (n^2+n+n*x)*P(n,x) = (8*n^2+12*n-8+(4*n^2+14*n-8)*x+(4*n-2)*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[((8*n^2 + 12*n - 8 + (4*n^2 + 14*n - 8)*x + (4*n - 2)*x^2)*P[n - 1, x])/(n^2 + n + 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] = ((8*n^2 + 12*n - 8 + (4*n^2 + 14*n - 8)*x + (4*n - 2)*x^2)*v[i - 1])/(n^2 + n + 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 basis_map_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
    • Geometric basis map (elementary) 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)
    • Geometric basis map (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)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Iterated real-rootedness preservers (above)
    • 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)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A120910 → D → D+1
• A093375 → x ↦ x+1 → reverse
• A178820 → basis A007318 → D+1
• A196347 → x ↦ x+1 → D+1
• A094305 → x ↦ x+1 → D+1
• A085881 → basis A007318 → D+1
• A038330 → basis A013609 → D+1
• A038207 → D+1
• A028326 → basis A007318 → D+1
• A085880 → x ↦ x+1 → D+1
• A093375 → D → basis A007318
• A120910 → D+1
• A178822 → x ↦ x+1 → D+1
• A156992 → basis A007318 → D+1
• A038208 → basis A007318 → D+1
• basis-map coefficients of A113214, shifted → x ↦ x+1 → D+1
• A135065 → x ↦ x+1 → D+1
• A038221 → x ↦ x+1 → D+1
• A178821 → basis A007318 → D+1
• A038338 → x ↦ x+1 → D+1
• A003506 → x ↦ x+1 → D+1
• A038325 → basis A007318 → D+1
• A093375 → x ↦ x+1
• A103406 → x ↦ x/(x+1)
• A027466 → basis A013611 → 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.