A038305 → θ+1

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Submission text ?

Triangle begins:
  0:          100,          120,            27;
  1:         1000,         1800,           810,           108;
  2:        10000,        24000,         16200,          4320,           405;
  3:       100000,       300000,        270000,        108000,         20250,         1458;
  4:      1000000,      3600000,       4050000,       2160000,        607500,        87480,         5103;
  5:     10000000,     42000000,      56700000,      37800000,      14175000,      3061800,       357210,       17496;
  6:    100000000,    480000000,     756000000,     604800000,     283500000,     81648000,     14288400,     1399680,      59049;
  7:   1000000000,   5400000000,    9720000000,    9072000000,    5103000000,   1837080000,    428652000,    62985600,    5314410,     196830;
  8:  10000000000,  60000000000,  121500000000,  129600000000,   85050000000,  36741600000,  10716300000,  2099520000,  265720500,   19683000,   649539;
  9: 100000000000, 660000000000, 1485000000000, 1782000000000, 1336500000000, 673596000000, 235758600000, 57736800000, 9743085000, 1082565000, 71449290, 2125764;

The row polynomials satisfy (10+(3*n+6)*x)*P(n,x) = (100+(30*n+120)*x+(9*n+27)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 120*x + 27*x^2.

P[0, x_] := 100 + 120*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((100 + (30*n + 120)*x + (9*n + 27)*x^2)*P[n - 1, x])/(10 + (3*n + 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 100 + 120*x + 27*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((100 + (30*n + 120)*x + (9*n + 27)*x^2)*v[i - 1])/(10 + (3*n + 6)*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 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 → 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.