A038321 → basis A007318 → D+1

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Triangle begins:
  0:             576,              350,               49;
  1:           12636,            12096,             3675,              343;
  2:          268272,           353808,           169344,            34300,             2401;
  3:         5563728,          9389520,          6191640,          1975680,           300125,           16807;
  4:       113374080,        233676576,        197179920,         86682960,         20744640,         2521050,          117649;
  5:      2278819008,       5555329920,       5725076112,       3220605360,       1061866260,       203297472,        20588575,         823543;
  6:     45304282368,     127613864448,     155549237760,     106868087424,      45088475040,     11892902112,      1897443072,      164708600,        5764801;
  7:    892616806656,    2854169789184,    4019836730112,    3266533992960,    1683172376928,    568114785504,    124875472176,    17076987648,     1297080225,      40353607;
  8:  17455617552384,   62483176465920,   99895942621440,   93796190369280,   57164344876800,  23564413276992,   6628005830880,  1248754721760,   149423641920,   10088401750,   282475249;
  9: 339194386529280, 1344082551533568, 2405602293937920, 2563995860616960, 1805576664608640, 880330911102720, 302409970388064, 72908064139680, 12019264196940, 1278402269760, 77680693475, 1977326743;

The row polynomials satisfy (7*n+25+7*x)*P(n,x) = (126*n+576+(49*n+350)*x+49*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 350*x + 49*x^2.

P[0, x_] := 576 + 350*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((126*n + 576 + (49*n + 350)*x + 49*x^2)*P[n - 1, x])/(7*n + 25 + 7*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 576 + 350*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((126*n + 576 + (49*n + 350)*x + 49*x^2)*v[i - 1])/(7*n + 25 + 7*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 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_A007318 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (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.