basis-map coefficients of A297191, shifted → (θ+1)² → basis A013610

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Triangle begins:
  0:   1;
  1:   9,    54,     81;
  2:  25,   300,   1350,    2700,     2025;
  3:  49,   882,   6615,   26460,    59535,     71442,      35721;
  4:  81,  1944,  20412,  122472,   459270,   1102248,    1653372,     1417176,       531441;
  5: 121,  3630,  49005,  392040,  2058210,   7409556,   18523890,    31755240,     35724645,     23816430,      7144929;
  6: 169,  6084, 100386, 1003860,  6776055,  32525064,  113837724,   292725576,    548860455,    731813940,    658632546,     359254116,      89813529;
  7: 225,  9450, 184275, 2211300, 18243225, 109459350,  492567075,  1688801400,   4433103675,   8866207350,  13299311025,   14508339300,   10881254475,    5022117450,    1076168025;
  8: 289, 13872, 312120, 4369680, 42604380, 306751536, 1687133448,  7230571920,  24403180230,  65075147280, 136657809288,  223621869744,  279527337180,  258025234320,  165873364920,   66349345968,   12440502369;
  9: 361, 19494, 497097, 7953552, 89477460, 751610664, 4885469316, 25125270768, 103641741918, 345472473060, 932775677262, 2035146932208, 3561507131364, 4931317566504, 5283554535540, 4226843628432, 2377599540993, 839152779174, 139858796529;

The row polynomials satisfy (4*n^2-4*n+1)*P(n,x) = (4*n^2+4*n+1+(24*n^2+24*n+6)*x+(36*n^2+36*n+9)*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[((4*n^2 + 4*n + 1 + (24*n^2 + 24*n + 6)*x + (36*n^2 + 36*n + 9)*x^2)*P[n - 1, x])/(4*n^2 - 4*n + 1)]];
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 + 4*n + 1 + (24*n^2 + 24*n + 6)*x + (36*n^2 + 36*n + 9)*x^2)*v[i - 1])/(4*n^2 - 4*n + 1)); 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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from theta_plus_one_squared of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of…?
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
        • Parent-row provers of polytool polytool ? step › direct check
        • 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 basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

      • from theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Del…?
        • Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

          • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from theta_plus_one of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Del…?
            • ↑ Laguerre’s theorem on multiplier sequences (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from theta_plus_one_squared of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of…?
        • ↑ Laguerre’s theorem on multiplier sequences (above)
        • ↑ Laguerre’s theorem on multiplier sequences (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A297191, shifted → basis A013610
• basis-map coefficients of A297191, shifted → basis A013610 → reverse
• A139548 → x ↦ x+1 → basis A007318
• basis-map coefficients of A297191, shifted → θ+1 → basis A013610
• A034870 → basis A013611 → x ↦ x/(x+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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.