A038321 → basis A013611 → D

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Submission text ?

Triangle begins:
  0:             1008,              1568;
  1:            27216,             84672,              65856;
  2:           653184,           3048192,            4741632,            2458624;
  3:         14696640,          91445760,          213373440,          221276160,            86051840;
  4:        317447424,        2469035520,         7681443840,        11948912640,          9293598720,          2891341824;
  5:       6666395904,       62219695104,       241965480960,       501854330880,        585496719360,        364309069824,         94450499584;
  6:     137137287168,     1493272682496,      6968605851648,     18066755911680,      28103842529280,      26230253027328,      13600871940096,       3022415986688;
  7:    2777030065152,    34558596366336,    188152357994496,    585362891538432,    1138205622435840,    1416433663475712,    1101670627147776,     489631389843456,      95206103580672;
  8:   55540601303040,   777568418242560,   4838203491287040,  17560886746152960,   40975402407690240,   63739514856407040,   66100237628866560,   44066825085911040,   17137098644520960,   2961967666954240;
  9: 1099703905800192, 17106505201336320, 119745536409354240, 496722225105469440, 1352188279453777920, 2524084788313718784, 3271961762628894720, 2908410455670128640, 1696572765807575040, 586469598056939520, 91228604142190592;

The row polynomials satisfy (n+1)*P(n,x) = (18*n+36+(28*n+56)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1008 + 1568*x.

P[0, x_] := 1008 + 1568*x;
P[n_, x_] := P[n, x] = Expand[Together[((18*n + 36 + (28*n + 56)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1008 + 1568*x; for(n = 1, N, my(i = n + 1); v[i] = ((18*n + 36 + (28*n + 56)*x)*v[i - 1])/(n + 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(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

Related ?

Same fingerprint ?

• A038303 → basis A013610 → basis A013610
• A038321 → basis A013611 → reverse
• A038281 → x ↦ x+1 → basis A013610
• A038321 → basis A013611

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.