A085880 → basis A013609 → basis A008277

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Submission text ?

Triangle begins:
   1:          1;
   2:          4,           2;
   3:         32,          40,            8;
   4:        320,         760,          360,           40;
   5:       3584,       14560,        12320,         3136,          224;
   6:      43008,      283584,       383040,       168000,        26880,         1344;
   7:     540672,     5617920,     11413248,      7687680,      2069760,       228096,        8448;
   8:    7028736,   113063808,    333260928,    324035712,    130690560,     23831808,     1921920,      54912;
   9:   93716480,  2308134400,   9640716800,  13022929920,   7387860480,   1983421440,   261381120,   16107520,    366080;
  10: 1274544128, 47723213824, 277922810880, 508336491520, 389980631040, 144648311808, 27706398720, 2763171840, 134424576, 2489344;

The row polynomials satisfy n*P(n,x) = (16*n-24+(8*n-12)*x)*P(n-1,x) + (8*n-12)*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n - 24 + (8*n - 12)*x)*P[n - 1, x] + (8*n - 12)*x*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n - 24 + (8*n - 12)*x)*v[i - 1] + (8*n - 12)*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, v[i])))};
T(10)

Properties

• Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
  by Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan nu…?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan nu…?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).?
            • 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) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • 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 the n-th Catalan number (A000108).?
                    • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                    • ↑ Liu–Wang theorem (above)
                    • ↑ Iterated real-rootedness preservers (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

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