A038325 → basis A007318 → basis A008277

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Submission text ?

Triangle begins:
  0:              1;
  1:             33,              11;
  2:           1089,             847,              121;
  3:          35937,           49247,            15972,             1331;
  4:        1185921,         2562175,          1420177,           263538,            14641;
  5:       39135393,       125780831,        106293660,         33015455,          4026275,          161051;
  6:     1291467969,      5964845887,       7229740441,       3348250290,        673193180,        58461513,        1771561;
  7:    42618442977,    276659366687,     463249029012,     300511663991,      88666628050,     12549738124,      818461182,      19487171;
  8:  1406408618241,  12641815006975,   28521949629217,   24929509142538,   10132958663751,   2079709863462,   219074776382,   11146661812,    214358881;
  9: 46411484401953, 571710355107551, 1707767194683660, 1959089049328895, 1054462417676745, 294477013285917, 44565211359900, 3635955339522, 148550704533, 2357947691;

The row polynomials satisfy P(n,x) = (33+11*x)*P(n-1,x) + 11*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(33 + 11*x)*P[n - 1, x] + 11*x*D[P[n - 1, x], 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] = (33 + 11*x)*v[i - 1] + 11*x*deriv(v[i - 1], x)); 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 Ray basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Ray basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Ray basis map theorem

Proof chains ? JSON

• Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ? step ›

  • from basis_map_A007318 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Iterated real-rootedness preservers (above)
            • Parent-row provers of polytool polytool ? step › direct check
        • ↑ Parent-row provers of polytool (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A027466 → basis A013611 → basis A008277
• A085881 → basis A007318 → basis A008277
• A094305 → x ↦ x+1 → basis A008277
• A196347 → x ↦ x+1 → basis A008277
• A143495: Triangle read by rows: 3-Stirling numbers of the second kind.
• A038330 → basis A013609 → basis A008277
• A038207 → basis A008277 → reverse
• A178820 → basis A007318 → basis A008277
• A038207 → basis A008277
• A120910 → D → basis A008277
• A120910 → basis A008277
• A038338 → x ↦ x+1 → basis A008277
• A178821 → basis A007318 → basis A008277
• A085880 → x ↦ x+1 → basis A008277
• A038221 → x ↦ x+1 → basis A008277
• A156992 → basis A007318 → basis A008277
• A003506 → x ↦ x+1 → basis A008277
• A038208 → basis A007318 → basis A008277
• A178822 → x ↦ x+1 → basis A008277
• A356546 → x ↦ x+1 → basis A008277
• A135065 → x ↦ x+1 → basis A008277
• A028326 → basis A007318 → basis A008277
• basis-map coefficients of A113214, shifted → x ↦ x+1 → basis A008277
• A143495 → 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. [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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.