A038333 → basis A013611 → x ↦ x+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:                2209,                 2632,                  784;
  1:              103823,               185556,               110544,                21952;
  2:             4879681,             11628176,             10391136,              4126976,                614656;
  3:           229345007,            683155340,            813972320,            484919680,             144444160,             17210368;
  4:         10779215329,          38529961176,          57385048560,          45582449920,           20366626560,           4853323776,            481890304;
  5:        506623120463,        2112726204484,        3775936195248,        3749156505920,         2233540046080,         798371761152,         158541910016,          13492928512;
  6:      23811286661761,      113483578983712,      236625334902208,      281936569245184,       209952764331520,      100062594064384,       29805879083008,        5073341120512,        377801998336;
  7:    1119130473102767,     6000444238763772,    14298930951947712,    19876528131785472,     17762003862446592,    10581619322308608,     4202628950704128,     1073011646988288,     159810245296128,      10578455953408;
  8:   52599132235830049,   313356532468774760,   840062193426928080,  1334566888848453120,   1391356969224983040,   994672216297009152,   493808901707735040,   168105158028165120,   37555407644590080,    4971874298101760,    296196766695424;
  9: 2472159215084012303, 16200532728635655092, 48256906000191313040, 86246385191831282880, 102761650441330890240, 85707589304258955264, 51059840436579803136, 21727591675140341760, 6472048584084357120, 1285229506059304960, 153133728381534208, 8293509467471872;

The row polynomials satisfy P(n,x) = (47+28*x)*P(n-1,x) for n >= 1, with P(0,x) = 2209 + 2632*x + 784*x^2.

P[0, x_] := 2209 + 2632*x + 784*x^2;
P[n_, x_] := P[n, x] = Expand[(47 + 28*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2209 + 2632*x + 784*x^2; for(n = 1, N, my(i = n + 1); v[i] = (47 + 28*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 Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (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)*12^(i-j)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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
        • 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_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
    • ↑ Geometric basis map (elementary) (above)

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.