A357367 → reverse

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Triangle begins:
  0:           1;
  1:           2;
  2:          12,           6;
  3:         120,         120,           24;
  4:        1680,        2520,         1080,          120;
  5:       30240,       60480,        40320,        10080,         720;
  6:      665280,     1663200,      1512000,       604800,      100800,        5040;
  7:    17297280,    51891840,     59875200,     33264000,     9072000,     1088640,      40320;
  8:   518918400,  1816214400,   2542700160,   1816214400,   698544000,   139708800,   12700800,    362880;
  9: 17643225600, 70572902400, 116237721600, 101708006400, 50854003200, 14529715200, 2235340800, 159667200, 3628800;

The row polynomials satisfy P(n,x) = (4*n-2+(3*n-3)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4*n - 2 + (3*n - 3)*x)*P[n - 1, x] - (2*x + 2*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; for(n = 1, N, my(i = n + 1); v[i] = (4*n - 2 + (3*n - 3)*x)*v[i - 1] - (2*x + 2*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 Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 - 1, k - 1)*(n + k)! / k!.?
    • 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
    • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

• A367270 → D → x ↦ x+1
• A386789: Triangle read by rows: T(n, k) = binomial(n - 1, k - 1)*binomial(n + k, k).
• A114656 → basis A007318 → basis A013609
• A132812 → x ↦ x/(x+1)
• A132812 → basis A007318
• narayana → basis A007318
• A033282: Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
• narayana → basis A007318 → reverse
• A063007 → D
• A386789 → reverse
• narayana → x ↦ x/(x+1)
• A132812 → basis A007318 → reverse
• A357367: Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.
• A126216: Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but …

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.