A106246 → basis A013611

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Submission text ?

Triangle begins:
  0:  1;
  1:  3,    4;
  2:  6,   24,    16;
  3: 10,   72,   144,     64;
  4: 15,  160,   576,    768,    256;
  5: 21,  300,  1600,   3840,   3840,    1024;
  6: 28,  504,  3600,  12800,  23040,   18432,    4096;
  7: 36,  784,  7056,  33600,  89600,  129024,   86016,   16384;
  8: 45, 1152, 12544,  75264, 268800,  573440,  688128,  393216,   65536;
  9: 55, 1620, 20736, 150528, 677376, 1935360, 3440640, 3538944, 1769472, 262144;

The row polynomials satisfy P(n,x) = (1+12*x+48*x^2+64*x^3)*P(n-3,x) - (3+24*x+48*x^2)*P(n-2,x) + (3+12*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 4*x, P(2,x) = 6 + 24*x + 16*x^2.

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[2, x_] := 6 + 24*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 12*x + 48*x^2 + 64*x^3)*P[n - 3, x] - (3 + 24*x + 48*x^2)*P[n - 2, x] + (3 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x; v[3] = 6 + 24*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = (1 + 12*x + 48*x^2 + 64*x^3)*v[i - 3] - (3 + 24*x + 48*x^2)*v[i - 2] + (3 + 12*x)*v[i - 1]); 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)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from Number triangle T(n,k)=C(n,k)C(2,n-k).?
    • Laguerre’s theorem on differential operators [SP14] polytool ? step ›

      • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
        • Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›

          • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
            • 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
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.?
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Veronese sections of Pólya frequency sequences (above)
            • ↑ Parent-row provers of polytool (above)
    • 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

References

1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
3. [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
4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.