A038305 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:          160,            78,             9;
  1:         1900,          1440,           351,            27;
  2:        22000,         22800,          8640,          1404,           81;
  3:       250000,        330000,        171000,         43200,         5265,          243;
  4:      2800000,       4500000,       2970000,       1026000,       194400,        18954,         729;
  5:     31000000,      58800000,      47250000,      20790000,      5386500,       816480,       66339,        2187;
  6:    340000000,     744000000,     705600000,     378000000,    124740000,     25855200,     3265920,      227448,       6561;
  7:   3700000000,    9180000000,   10044000000,    6350400000,   2551500000,    673596000,   116348400,    12597120,     767637,     19683;
  8:  40000000000,  111000000000,  137700000000,  100440000000,  47628000000,  15309000000,  3367980000,   498636000,   47239200,   2558790,   59049;
  9: 430000000000, 1320000000000, 1831500000000, 1514700000000, 828630000000, 314344800000, 84199500000, 15877620000, 2056873500, 173210400, 8444007, 177147;

The row polynomials satisfy (3*n+13+3*x)*P(n,x) = (30*n+160+(9*n+78)*x+9*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 160 + 78*x + 9*x^2.

P[0, x_] := 160 + 78*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((30*n + 160 + (9*n + 78)*x + 9*x^2)*P[n - 1, x])/(3*n + 13 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 160 + 78*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((30*n + 160 + (9*n + 78)*x + 9*x^2)*v[i - 1])/(3*n + 13 + 3*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 Triangle whose (i,j)-th entry is binomial(i,j)*10^(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 ?

• A038291 → basis A013610 → D+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.