A156992 → basis A013609 → basis A008277

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Triangle begins:
  0:            1;
  1:            8,              4;
  2:           96,            120,              24;
  3:         1536,           3648,            1728,             192;
  4:        30720,         124800,          105600,           26880,            1920;
  5:       737280,        4861440,         6566400,         2880000,          460800,           23040;
  6:     20643840,      214502400,       435778560,       293529600,        79027200,         8709120,         322560;
  7:    660602880,    10626416640,     31321866240,     30454824960,     12283084800,      2239856640,      180633600,       5160960;
  8:  23781703680,   585717350400,   2446449868800,   3304727838720,   1874760007680,    503317463040,    66328657920,    4087480320,     92897280;
  9: 951268147200, 35618675097600, 207430336512000, 379401781248000, 291065757696000, 107959644979200, 20678934528000, 2062319616000, 100329062400, 1857945600;

The row polynomials satisfy P(n,x) = (4*n+4+(2*n+2)*x)*P(n-1,x) + (2*n+2)*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[(4*n + 4 + (2*n + 2)*x)*P[n - 1, x] + (2*n + 2)*x*D[P[n - 1, x], 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] = (4*n + 4 + (2*n + 2)*x)*v[i - 1] + (2*n + 2)*x*deriv(v[i - 1], 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 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_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
            • 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
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • Geometric basis map (elementary) polytool ? step ›

                  • from theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
                    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

                      • from Triangle read by rows: row n consists of n zeros followed by n!.?
                        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
                        • Parent-row provers of polytool polytool ? step › direct check
                        • Parent-row provers of polytool polytool ? step › direct check
                        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                          • from Triangle read by rows: row n consists of n zeros followed by n!.?
                            • ↑ Iterated real-rootedness preservers (above)
                            • ↑ Parent-row provers of polytool (above)
                            • ↑ Parent-row provers of polytool (above)
                            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                              • from Triangle read by rows: row n consists of n zeros followed by n!.?
                                • ↑ Iterated real-rootedness preservers (above)
                                • ↑ Parent-row provers of polytool (above)
                                • ↑ Parent-row provers of polytool (above)
                        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

                      • from Triangle read by rows: row n consists of n zeros followed by n!.?
                        • ↑ Iterated real-rootedness preservers (above)
                        • ↑ Parent-row provers of polytool (above)
                        • ↑ Parent-row provers of polytool (above)
                        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
                    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                      • from theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
                        • ↑ Laguerre’s theorem on multiplier sequences (above)
                        • ↑ Laguerre’s theorem on multiplier sequences (above)
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
                • Parent-row provers of polytool polytool ? step › direct check
                • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

                  • from theta_plus_one of Triangle read by rows: row n consists of n zeros followed by n!.?
                    • ↑ Laguerre’s theorem on multiplier sequences (above)
                    • ↑ Laguerre’s theorem on multiplier sequences (above)
                    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Geometric basis map (elementary) (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A376827 → basis A008277
• A028326 → D → basis A008277
• A003506 → D → basis A008277
• basis-map coefficients of A127647 → basis A008277
• A085880 → basis A013609 → basis A008277
• A038208 → basis A008277
• A178820 → basis A008277
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
• A124860 → basis A008277
• A027465 → basis A013611 → basis A008277
• A178821 → basis A008277
• A094305 → basis A008277
• A178820 → basis A013609 → basis A008277
• A156992 → basis A008277
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficient…
• A085881 → basis A008277
• A356546 → basis A013609 → basis A008277
• A156992 → basis A008277 → reverse
• A038338 → basis A008277
• A038338 → basis A013609 → basis A008277
• A135089 → basis A008277
• A038325 → basis A013609 → basis A008277
• basis-map coefficients of A113214, shifted → basis A008277
• A196347 → basis A013609 → basis A008277
• A038208 → D → basis A008277
• A344565 → basis A008277
• A028326 → basis A013609 → basis A008277
• A038221 → basis A013609 → basis A008277
• A143494 → reverse
• A134058 → basis A008277
• A356546 → basis A008277
• A085880 → D → basis A008277
• basis-map coefficients of A112227 → x ↦ x+1 → basis A008277
• A178822 → basis A013609 → basis A008277
• A178822 → basis A008277
• basis-map coefficients of A026729, shifted → basis A007318 → basis A008277
• A028326 → basis A008277
• A134059 → basis A008277
• A038221 → basis A008277
• A132200 → basis A008277
• A135065 → basis A008277
• A196347 → basis A008277
• A135838 → basis A008277
• A131431 → x ↦ x+1 → basis A008277
• A258758 → basis A008277
• A178821 → basis A013609 → basis A008277
• A085881 → basis A013609 → basis A008277
• A120910 → basis A013610 → basis A008277
• basis-map coefficients of A113214, shifted → basis A013609 → basis A008277
• A038325 → basis A008277
• basis-map coefficients of A021012 → basis A008277 → D+1
• A126351 → reverse
• A085880 → basis A008277
• A038330 → basis A013611 → basis A008277
• A134346 → basis A008277
• A003506 → basis A008277
• basis-map coefficients of A113214, shifted → D → 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.