A038331 → θ+1

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Submission text ?

Triangle begins:
  0:          144,           240,             75;
  1:         1728,          4320,           2700,            500;
  2:        20736,         69120,          64800,          24000,           3125;
  3:       248832,       1036800,        1296000,         720000,         187500,          18750;
  4:      2985984,      14929920,       23328000,       17280000,        6750000,        1350000,         109375;
  5:     35831808,     209018880,      391910400,      362880000,      189000000,       56700000,        9187500,        625000;
  6:    429981696,    2866544640,     6270566400,     6967296000,     4536000000,     1814400000,      441000000,      60000000,       3515625;
  7:   5159780352,   38698352640,    96745881600,   125411328000,    97977600000,    48988800000,    15876000000,    3240000000,     379687500,     19531250;
  8:  61917364224,  515978035200,  1451188224000,  2149908480000,  1959552000000,  1175731200000,   476280000000,  129600000000,   22781250000,   2343750000,   107421875;
  9: 743008370688, 6810910064640, 21284093952000, 35473489920000, 36951552000000, 25866086400000, 12573792000000, 4276800000000, 1002375000000, 154687500000, 14179687500, 585937500;

The row polynomials satisfy (12+(5*n+10)*x)*P(n,x) = (144+(60*n+240)*x+(25*n+75)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 240*x + 75*x^2.

P[0, x_] := 144 + 240*x + 75*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((144 + (60*n + 240)*x + (25*n + 75)*x^2)*P[n - 1, x])/(12 + (5*n + 10)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 240*x + 75*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((144 + (60*n + 240)*x + (25*n + 75)*x^2)*v[i - 1])/(12 + (5*n + 10)*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)*12^(i-j)*5^j.?
    • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    • 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
    • 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

Related ?

Same fingerprint ?

• A038271 → 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.