basis-map coefficients of A297191, shifted → (θ+1)² → basis A013609

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Triangle begins:
  0:   1;
  1:   9,    36,     36;
  2:  25,   200,    600,     800,      400;
  3:  49,   588,   2940,    7840,    11760,     9408,      3136;
  4:  81,  1296,   9072,   36288,    90720,   145152,    145152,      82944,      20736;
  5: 121,  2420,  21780,  116160,   406560,   975744,   1626240,    1858560,    1393920,     619520,      123904;
  6: 169,  4056,  44616,  297440,  1338480,  4283136,   9993984,   17132544,   21415680,   19036160,    11421696,     4153344,      692224;
  7: 225,  6300,  81900,  655200,  3603600, 14414400,  43243200,   98841600,  172972800,  230630400,   230630400,   167731200,    83865600,    25804800,     3686400;
  8: 289,  9248, 138720, 1294720,  8415680, 40395264, 148115968,  423188480,  952174080, 1692753920,  2369855488,  2585296896,  2154414080,  1325793280,   568197120,  151519232,   18939904;
  9: 361, 12996, 220932, 2356608, 17674560, 98977536, 428902656, 1470523392, 4043939328, 8986531840, 16175757312, 23528374272, 27449769984, 25338249216, 18098749440, 9652666368, 3619749888, 851705856, 94633984;

The row polynomials satisfy (4*n^2-4*n+1)*P(n,x) = (4*n^2+4*n+1+(16*n^2+16*n+4)*x+(16*n^2+16*n+4)*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 + (16*n^2 + 16*n + 4)*x + (16*n^2 + 16*n + 4)*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 + (16*n^2 + 16*n + 4)*x + (16*n^2 + 16*n + 4)*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 ?

• A034870 → x ↦ x/(x+1)
• A034870 → basis A013611 → basis A013610
• A034870 → basis A013609 → x ↦ x+1
• basis-map coefficients of A297191, shifted → θ+1 → basis A013609
• A139548 → basis A013609 → basis A007318
• A139548 → x ↦ x/(x+1)
• basis-map coefficients of A297191, shifted → basis A013609
• A034870 → basis A013611
• A034870 → basis A013609 → basis A013611
• A139548 → x ↦ x+1 → reverse
• A034870 → x ↦ x+1
• A139548 → basis A013611
• A139548 → x ↦ x+1
• basis-map coefficients of A297191, shifted → basis A013609 → basis A013610

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.