A038334 → θ+1

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No OEIS matches.

Submission text ?

Triangle begins:
  0:          144,            384,            192;
  1:         1728,           6912,           6912,            2048;
  2:        20736,         110592,         165888,           98304,           20480;
  3:       248832,        1658880,        3317760,         2949120,         1228800,          196608;
  4:      2985984,       23887872,       59719680,        70778880,        44236800,        14155776,         1835008;
  5:     35831808,      334430208,     1003290624,      1486356480,      1238630400,       594542592,       154140672,        16777216;
  6:    429981696,     4586471424,    16052649984,     28538044416,     29727129600,     19025362944,      7398752256,      1610612736,      150994944;
  7:   5159780352,    61917364224,   247669456896,    513684799488,    642105999360,    513684799488,    266355081216,     86973087744,    16307453952,     1342177280;
  8:  61917364224,   825564856320,  3715041853440,   8806025134080,  12842119987200,  12328435187712,   7990652436480,   3478923509760,   978447237120,   161061273600,   11811160064;
  9: 743008370688, 10897456103424, 54487280517120, 145299414712320, 242165691187200, 271225574129664, 210953224323072, 114804475822080, 43051678433280, 10630044057600, 1559073128448, 103079215104;

The row polynomials satisfy (3+(2*n+4)*x)*P(n,x) = (36+(24*n+96)*x+(16*n+48)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 384*x + 192*x^2.

P[0, x_] := 144 + 384*x + 192*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((36 + (24*n + 96)*x + (16*n + 48)*x^2)*P[n - 1, x])/(3 + (2*n + 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 384*x + 192*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((36 + (24*n + 96)*x + (16*n + 48)*x^2)*v[i - 1])/(3 + (2*n + 4)*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)*8^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 ?

• A038220 → θ+1
• A120910 → basis A013609 → θ+1
• A038243 → basis A013611 → 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.