A038330 → θ+1

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Submission text ?

Triangle begins:
  0:          144,           192,             48;
  1:         1728,          3456,           1728,            256;
  2:        20736,         55296,          41472,          12288,           1280;
  3:       248832,        829440,         829440,         368640,          76800,          6144;
  4:      2985984,      11943936,       14929920,        8847360,        2764800,        442368,         28672;
  5:     35831808,     167215104,      250822656,      185794560,       77414400,      18579456,       2408448,       131072;
  6:    429981696,    2293235712,     4013162496,     3567255552,     1857945600,     594542592,     115605504,     12582912,       589824;
  7:   5159780352,   30958682112,    61917364224,    64210599936,    40131624960,   16052649984,    4161798144,    679477248,     63700992,     2621440;
  8:  61917364224,  412782428160,   928760463360,  1100753141760,   802632499200,  385263599616,  124853944320,  27179089920,   3822059520,   314572800,   11534336;
  9: 743008370688, 5448728051712, 13621820129280, 18162426839040, 15135355699200, 8475799191552, 3296144130048, 896909967360, 168170618880, 20761804800, 1522532352, 50331648;

The row polynomials satisfy (3+(n+2)*x)*P(n,x) = (36+(12*n+48)*x+(4*n+12)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 192*x + 48*x^2.

P[0, x_] := 144 + 192*x + 48*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((36 + (12*n + 48)*x + (4*n + 12)*x^2)*P[n - 1, x])/(3 + (n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 192*x + 48*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((36 + (12*n + 48)*x + (4*n + 12)*x^2)*v[i - 1])/(3 + (n + 2)*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)*4^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 ?

• image under θ+1 → θ+1 → D+1 → basis A013610
• A027465 → D → θ+1
• A027465 → θ+1
• A038279 → basis A013610 → θ+1
• A131431 → D+1 → basis A013610
• A120910 → basis A007318 → θ+1
• A038243 → basis A013609 → θ+1
• A132440 → D+1 → basis A013610

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.