A038329 → basis A013611 → θ+1

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Submission text ?

Triangle begins:
  0:           225,             720,             432;
  1:          3375,           16200,           19440,             6912;
  2:         50625,          324000,          583200,           414720,           103680;
  3:        759375,         6075000,        14580000,         15552000,          7776000,          1492992;
  4:      11390625,       109350000,       328050000,        466560000,        349920000,        134369280,         20901888;
  5:     170859375,      1913625000,      6889050000,      12247200000,      12247200000,       7054387200,       2194698240,        286654464;
  6:    2562890625,     32805000000,    137781000000,     293932800000,     367416000000,     282175488000,     131681894400,      34398535680,       3869835264;
  7:   38443359375,    553584375000,   2657205000000,    6613488000000,    9920232000000,    9523422720000,    5925685248000,    2321901158400,     522427760640,     51597803520;
  8:  576650390625,   9226406250000,  49822593750000,  141717600000000,  248005800000000,  285702681600000,  222213196800000,  116095057920000,   39182082048000,   7739670528000,    681091006464;
  9: 8649755859375, 152235703125000, 913414218750000, 2922925500000000, 5845851000000000, 7856823744000000, 7333035494400000, 4788921139200000, 2155014512640000, 638522818560000, 112380016066560, 8916100448256;

The row polynomials satisfy (5+(4*n+8)*x)*P(n,x) = (75+(60*n+240)*x+(48*n+144)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 720*x + 432*x^2.

P[0, x_] := 225 + 720*x + 432*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((75 + (60*n + 240)*x + (48*n + 144)*x^2)*P[n - 1, x])/(5 + (4*n + 8)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 225 + 720*x + 432*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((75 + (60*n + 240)*x + (48*n + 144)*x^2)*v[i - 1])/(5 + (4*n + 8)*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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›

  • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^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

Related ?

Same fingerprint ?

• A038334 → basis A013609 → θ+1
• A013611 → basis A007318 → θ+1

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.