A105278 → reverse

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Submission text ?

Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  6,    6;
   4: 1, 12,   36,    24;
   5: 1, 20,  120,   240,    120;
   6: 1, 30,  300,  1200,   1800,     720;
   7: 1, 42,  630,  4200,  12600,   15120,     5040;
   8: 1, 56, 1176, 11760,  58800,  141120,   141120,    40320;
   9: 1, 72, 2016, 28224, 211680,  846720,  1693440,  1451520,   362880;
  10: 1, 90, 3240, 60480, 635040, 3810240, 12700800, 21772800, 16329600, 3628800;

The row polynomials satisfy P(n,x) = (1+(2*n-2)*x+(n^2-3*n+2)*x^2)*P(n-1,x) - (2*x^2+(2*n-4)*x^3)*d/dx P(n-1,x) + x^4*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n - 2)*x + (n^2 - 3*n + 2)*x^2)*P[n - 1, x] - (2*x^2 + (2*n - 4)*x^3)*D[P[n - 1, x], x] + x^4*D[P[n - 1, x], {x, 2}]];
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] = (1 + (2*n - 2)*x + (n^2 - 3*n + 2)*x^2)*v[i - 1] - (2*x^2 + (2*n - 4)*x^3)*deriv(v[i - 1], x) + x^4*deriv(deriv(v[i - 1], x), x)); 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 Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.?
    • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • 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

Related ?

Same fingerprint ?

• A089231: Triangular array A066667 or A008297 unsigned and transposed.
• A089231 → D
• A130534 → basis A008277
• A225470 → basis A013610 → basis A008277
• A271703 → reverse
• A225471 → basis A013611 → basis A008277
• A271703: Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0)…
• A132393 → basis A008277 → D
• A105278: Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.
• A028338 → 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.