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

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Submission text ?

Triangle begins:
  0:  1;
  1:  3,  12,    12;
  2:  5,  40,   120,    160,     80;
  3:  7,  84,   420,   1120,   1680,    1344,      448;
  4:  9, 144,  1008,   4032,  10080,   16128,    16128,     9216,      2304;
  5: 11, 220,  1980,  10560,  36960,   88704,   147840,   168960,    126720,     56320,     11264;
  6: 13, 312,  3432,  22880, 102960,  329472,   768768,  1317888,   1647360,   1464320,    878592,     319488,      53248;
  7: 15, 420,  5460,  43680, 240240,  960960,  2882880,  6589440,  11531520,  15375360,  15375360,   11182080,    5591040,    1720320,    245760;
  8: 17, 544,  8160,  76160, 495040, 2376192,  8712704, 24893440,  56010240,  99573760, 139403264,  152076288,  126730240,   77987840,  33423360,   8912896,   1114112;
  9: 19, 684, 11628, 124032, 930240, 5209344, 22573824, 77395968, 212838912, 472975360, 851355648, 1238335488, 1444724736, 1333592064, 952565760, 508035072, 190513152, 44826624, 4980736;

The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+(8*n+4)*x+(8*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[((2*n + 1 + (8*n + 4)*x + (8*n + 4)*x^2)*P[n - 1, x])/(2*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] = ((2*n + 1 + (8*n + 4)*x + (8*n + 4)*x^2)*v[i - 1])/(2*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 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 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)
    • 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)

Related ?

Same fingerprint ?

• A034870 → basis A013609 → basis A013611
• basis-map coefficients of A297191, shifted → (θ+1)² → basis A013609
• basis-map coefficients of A297191, shifted → basis A013609
• A034870 → basis A013611
• A034870 → x ↦ x+1
• A034870 → basis A013609 → x ↦ x+1
• A139548 → x ↦ x+1 → reverse
• A139548 → basis A013611
• A034870 → basis A013611 → basis A013610
• basis-map coefficients of A297191, shifted → basis A013609 → basis A013610
• A034870 → x ↦ x/(x+1)
• A139548 → x ↦ x+1
• A139548 → x ↦ x/(x+1)
• A139548 → basis A013609 → basis A007318

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.