A269944 → basis A013609 → (θ+1)²

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Triangle begins:
  0:          1;
  1:          1,           8;
  2:          2,          24,           36;
  3:         10,         136,          288,          128;
  4:        100,        1440,         3492,         2304,          400;
  5:       1700,       25280,        65844,        51584,        14000,        1152;
  6:      44200,      670880,      1825704,      1575296,       525200,       70272,       3136;
  7:    1635400,    25176160,     70570008,     64777344,     24355200,     4112640,     307328,      8192;
  8:   81770000,  1271891200,   3641793120,   3489782784,   1420189200,   275774976,   26561920,   1212416,   20736;
  9: 5315050000, 83327088000, 242440063200, 239784478720, 103217869200, 22015518336, 2477245568, 148193280, 4416768, 51200;

The row polynomials satisfy P(n,x) = -(n^4-8*n^3+26*n^2-40*n+25+(8*n^2-32*n+40)*x-36*x^2)*P(n-2,x) + ((8*n^2-32*n+40)*x^2+16*x^3)*d/dx P(n-2,x) + (2*n^2-6*n+7+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.

P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (8*n^2 - 32*n + 40)*x - 36*x^2)*P[n - 2, x] + ((8*n^2 - 32*n + 40)*x^2 + 16*x^3)*D[P[n - 2, x], x] + (2*n^2 - 6*n + 7 + 8*x)*P[n - 1, x] - 8*x^2*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; v[2] = 1 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (8*n^2 - 32*n + 40)*x - 36*x^2)*v[i - 2] + ((8*n^2 - 32*n + 40)*x^2 + 16*x^3)*deriv(v[i - 2], x) + (2*n^2 - 6*n + 7 + 8*x)*v[i - 1] - 8*x^2*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 Laguerre’s theorem on multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A013609 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k >…?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…?
        • 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
        • Parent-row provers of polytool polytool ? step › direct check
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k)…?
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

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.