A038330 → basis A013610 → basis A007318

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:              784,               672,               144;
  1:            21952,             28224,             12096,               1728;
  2:           614656,           1053696,            677376,             193536,             20736;
  3:         17210368,          36879360,          31610880,           13547520,           2903040,            248832;
  4:        481890304,        1239146496,        1327656960,          758661120,         243855360,          41803776,           2985984;
  5:      13492928512,       40478785536,       52044152832,        37174394880,       15931883520,        4096770048,         585252864,         35831808;
  6:     377801998336,     1295321137152,     1942981705728,      1665412890624,      892185477120,      305892163584,       65548320768,       8026324992,        429981696;
  7:   10578455953408,    40802615820288,    69947341406208,     69947341406208,    44966148046848,    19271206305792,     5506058944512,    1011316948992,     108355387392,      5159780352;
  8:  296196766695424,  1269414714408960,  2448156949217280,   2797893656248320,  2098420242186240,  1079187553124352,   385424126115840,   94389581905920,   15169754234880,   1444738498560,    61917364224;
  9: 8293509467471872, 39097973203795968, 83781371150991360, 107718905765560320, 92330490656194560, 55398294393716736, 23742126168735744, 7267997806755840, 1557428101447680, 222489728778240, 19070548180992, 743008370688;

The row polynomials satisfy P(n,x) = (28+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 784 + 672*x + 144*x^2.

P[0, x_] := 784 + 672*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(28 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 784 + 672*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (28 + 12*x)*v[i - 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 (−∞, -1]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • Geometric basis map (elementary) 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
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038335 → x ↦ x+1 → reverse
• A038255 → basis A013610
• A013609 → basis A013609 → x ↦ x/(x+1)
• A038255 → basis A013610 → reverse
• A038284 → basis A007318
• A038335 → x ↦ x+1 → D
• A038284 → basis A007318 → reverse
• A120910 → basis A013611 → x ↦ x/(x+1)
• A038255 → basis A013610 → D
• A038284 → basis A007318 → D
• A038303 → basis A013610 → basis A013609
• A038335 → 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.