A167194 → basis A007318

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Submission text ?

Triangle begins:
  0: 1;
  1: 3,  1;
  2: 4,  4,   1;
  3: 4,  8,   5,   1;
  4: 4, 12,  13,   6,   1;
  5: 4, 16,  25,  19,   7,   1;
  6: 4, 20,  41,  44,  26,   8,   1;
  7: 4, 24,  61,  85,  70,  34,   9,  1;
  8: 4, 28,  85, 146, 155, 104,  43, 10,  1;
  9: 4, 32, 113, 231, 301, 259, 147, 53, 11, 1;

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

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[2, x_] := 4 + 4*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*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] = 3 + x; v[3] = 4 + 4*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*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 ≥ 3
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from Triangle read by rows. A130713 in the columns.?
    • 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
    • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

• A159854 → basis A013609 → reverse
• A159854 → basis A013609
• A391533: The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
• A391533 → reverse

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.