A127952 → basis A013609

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Submission text ?

Triangle begins:
  0:    1;
  1:    2,     4;
  2:    6,    18,     12;
  3:   16,    64,     80,     32;
  4:   40,   200,    360,    280,     80;
  5:   96,   576,   1344,   1536,    864,    192;
  6:  224,  1568,   4480,   6720,   5600,   2464,    448;
  7:  512,  4096,  13824,  25600,  28160,  18432,   6656,   1024;
  8: 1152, 10368,  40320,  88704, 120960, 104832,  56448,  17280,  2304;
  9: 2560, 25600, 112640, 286720, 465920, 501760, 358400, 163840, 43520, 5120;

The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(8*n-16)*x+(4*n-8)*x^2)*P(n-2,x) + (4*n-8+(4*n-8)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 6 + 18*x + 12*x^2.

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 6 + 18*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (8*n - 16)*x + (4*n - 8)*x^2)*P[n - 2, x] + (4*n - 8 + (4*n - 8)*x)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; v[3] = 6 + 18*x + 12*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (8*n - 16)*x + (4*n - 8)*x^2)*v[i - 2] + (4*n - 8 + (4*n - 8)*x)*v[i - 1])/(n - 2)); 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)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from Triangle read by rows, T(n,k) = (n+1)*binomial(n-1,k-1).?
    • 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 ?

• A134239: A127899(unsigned) * A007318.
• A097805 → basis A013609
• A007318 → Pₙ + Pₙ₊₁
• A136158 → basis A013611
• A029635: The (1,2)-Pascal triangle (or Lucas triangle) read by rows.
• A093560 → basis A013609
• A028326 → Pₙ + Pₙ₊₁
• A128540 → basis A007318
• A029653: Numbers in (2,1)-Pascal triangle (by row).

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.