A038271 → basis A013611 → basis A007318

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Submission text ?

Triangle begins:
  0:              1;
  1:             32,              20;
  2:           1024,            1280,             400;
  3:          32768,           61440,           38400,            8000;
  4:        1048576,         2621440,         2457600,         1024000,          160000;
  5:       33554432,       104857600,       131072000,        81920000,        25600000,         3200000;
  6:     1073741824,      4026531840,      6291456000,      5242880000,      2457600000,       614400000,        64000000;
  7:    34359738368,    150323855360,    281857228800,    293601280000,    183500800000,     68812800000,     14336000000,     1280000000;
  8:  1099511627776,   5497558138880,  12025908428800,  15032385536000,  11744051200000,   5872025600000,   1835008000000,   327680000000,   25600000000;
  9: 35184372088832, 197912092999680, 494780232499200, 721554505728000, 676457349120000, 422785843200000, 176160768000000, 47185920000000, 7372800000000, 512000000000;

The row polynomials satisfy P(n,x) = (32+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(32 + 20*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; for(n = 1, N, my(i = n + 1); v[i] = (32 + 20*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 (−∞, -1]
  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 Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^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
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038334 → basis A013611
• basis-map coefficients of A112227 → basis A013611 → basis A013609
• A136572 → basis A013611 → basis A013609
• A038329 → basis A013610 → x ↦ x/(x+1)
• A013622 → basis A007318
• A120910 → basis A013609 → basis A013611
• A038336 → x ↦ x+1 → basis A013609
• A038259 → basis A007318 → basis A013609
• A038319 → basis A013609
• A013611 → D → basis A013609
• basis-map coefficients of A026729 → basis A013611 → basis A013609
• A013622 → basis A007318 → D
• A038220 → basis A013611
• A038220 → basis A013611 → D
• A130123 → basis A013611 → basis A013609
• A038319 → basis A013609 → reverse
• A038319 → basis A013609 → D

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.