A038329 → basis A013609 → basis A013609

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Submission text ?

Triangle begins:
  0:             441,              504,              144;
  1:            9261,            15876,             9072,              1728;
  2:          194481,           444528,           381024,            145152,             20736;
  3:         4084101,         11668860,         13335840,           7620480,           2177280,           248832;
  4:        85766121,        294055272,        420078960,         320060160,         137168640,         31352832,          2985984;
  5:      1801088541,       7204354164,      12350321424,       11762210880,        6721263360,       2304433152,        438939648,         35831808;
  6:     37822859361,     172904499936,     345808999872,      395210285568,      282293061120,     129048256512,      36870930432,       6019743744,       429981696;
  7:    794280046581,    4084868810988,    9336842996544,    12449123995392,    10670677710336,    6097530120192,    2322868617216,     568865783808,     81266540544,      5159780352;
  8:  16679880978201,   95313605589720,  245092128659280,   373473719861760,   373473719861760,  256096265048064,  121950602403840,   39820604866560,   8532986757120,   1083553873920,    61917364224;
  9: 350277500542221, 2201744289122532, 6290697968921520, 10784053661008320, 12324632755438080, 9859706204350464, 5634117831057408, 2299639931043840, 657039980298240, 125150472437760, 14302911135744, 743008370688;

The row polynomials satisfy P(n,x) = (21+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 441 + 504*x + 144*x^2.

P[0, x_] := 441 + 504*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(21 + 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] = 441 + 504*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 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 basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle of coefficients in expansion of (1+5x)^n.?
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
            • 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_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
            • Geometric basis map (elementary) polytool ? step ›

              • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
                • 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 [LW07, Thm 2.1, Cor 2.4] ? step › direct check
                • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
                • Parent-row provers of polytool polytool ? step › direct check
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038328 → basis A013611 → D
• A038334 → x ↦ x+1 → basis A013609
• A038255 → basis A013611 → reverse
• A013610 → basis A007318 → x ↦ x/(x+1)
• A038255 → basis A013611
• A120910 → basis A013611 → x ↦ x+1
• A038328 → basis A013611 → reverse
• A038335 → x ↦ x/(x+1)
• A038328 → basis A013611
• A038255 → basis A013611 → D
• A038291 → basis A013611 → basis A013609
• A038284 → x ↦ x/(x+1)
• A038330 → basis A013610 → x ↦ x/(x+1)

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. [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
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.