A356546 → basis A013611 → basis A008277

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Submission text ?

Triangle begins:
  0:            1;
  1:           12,              8;
  2:          216,            384,              96;
  3:         4320,          15680,            9600,             1280;
  4:        90720,         609280,          689920,           215040,            17920;
  5:      1959552,       23240448,        43545600,         23869440,          4515840,          258048;
  6:     43110144,      880889856,      2576200704,       2214051840,        714362880,        90832896,         3784704;
  7:    960740352,    33359259648,    147049943040,     186300162048,      91268136960,     19385253888,      1771241472,       56229888;
  8:  21616657920,  1265383342080,   8220510904320,   14781010083840,   10324123729920,   3259084308480,    490043473920,    33737932800,    843448320;
  9: 489977579520, 48130022420480, 453895036108800, 1129329739366400, 1081406880153600, 476123483996160, 104786645483520, 11738551418880, 630899343360, 12745441280;

The row polynomials satisfy n*P(n,x) = (24*n-12+(16*n-8)*x)*P(n-1,x) + (16*n-8)*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((24*n - 12 + (16*n - 8)*x)*P[n - 1, x] + (16*n - 8)*x*D[P[n - 1, x], x])/(n)]];
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] = ((24*n - 12 + (16*n - 8)*x)*v[i - 1] + (16*n - 8)*x*deriv(v[i - 1], x))/(n)); 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 Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from basis_map_A013611 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 basis_map_A013609 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013609 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 (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                • ↑ 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)
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A013609 → D → basis A008277
• A136572 → basis A013609 → basis A008277
• A085881 → basis A013611 → basis A008277
• A156992 → basis A013611 → basis A008277
• A038221 → basis A013611 → basis A008277
• A178822 → basis A013611 → basis A008277
• A120909 → basis A013610 → basis A008277
• A028326 → basis A013611 → basis A008277
• A038208 → basis A013611 → basis A008277
• A196347 → basis A013611 → basis A008277
• A130123 → basis A013609 → basis A008277
• A120909 → basis A008277
• basis-map coefficients of A026729 → basis A013609 → basis A008277
• basis-map coefficients of A112227 → basis A013609 → basis A008277
• basis-map coefficients of A030528, shifted → basis A013609 → basis A008277
• A178821 → basis A013611 → basis A008277
• A085880 → basis A013611 → basis A008277
• basis-map coefficients of A026729, shifted → basis A013609 → basis A008277
• A013609 → basis A013610 → basis A008277
• A178820 → basis A013611 → basis A008277
• A013609 → basis A008277

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.