A038305 → D

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:           60,           18;
  1:          900,          540,            81;
  2:        12000,        10800,          3240,           324;
  3:       150000,       180000,         81000,         16200,         1215;
  4:      1800000,      2700000,       1620000,        486000,        72900,         4374;
  5:     21000000,     37800000,      28350000,      11340000,      2551500,       306180,       15309;
  6:    240000000,    504000000,     453600000,     226800000,     68040000,     12247200,     1224720,      52488;
  7:   2700000000,   6480000000,    6804000000,    4082400000,   1530900000,    367416000,    55112400,    4723920,    177147;
  8:  30000000000,  81000000000,   97200000000,   68040000000,  30618000000,   9185400000,  1837080000,  236196000,  17714700,   590490;
  9: 330000000000, 990000000000, 1336500000000, 1069200000000, 561330000000, 202078800000, 50519700000, 8660520000, 974308500, 64953900, 1948617;

The row polynomials satisfy (n+1)*P(n,x) = (10*n+20+(3*n+6)*x)*P(n-1,x) for n >= 1, with P(0,x) = 60 + 18*x.

P[0, x_] := 60 + 18*x;
P[n_, x_] := P[n, x] = Expand[Together[((10*n + 20 + (3*n + 6)*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] = 60 + 18*x; for(n = 1, N, my(i = n + 1); v[i] = ((10*n + 20 + (3*n + 6)*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 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
• A038305 → reverse
• A038291 → basis A013610
• A038291 → basis A013610 → reverse
• A013612 → basis A013611 → D
• A038335 → x ↦ x+1 → basis A007318
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
• A013612 → basis A013611
• A038284 → basis A007318 → x ↦ x+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.