A038328 → x ↦ x+1 → D+1

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Submission text ?

Triangle begins:
  0:            252,             64,              4;
  1:           3920,           1512,            192,             8;
  2:          60368,          31360,           6048,           512,            16;
  3:         921984,         603680,         156800,         20160,          1280,           32;
  4:       13983424,       11063808,        3622080,        627200,         60480,         3072,          64;
  5:      210827008,      195767936,       77446656,      16903040,       2195200,       169344,        7168,        128;
  6:     3162405120,     3373232128,     1566143488,     413048832,      67612160,      7024640,      451584,      16384,       256;
  7:    47225249792,    56923292160,    30359089152,    9396860928,    1858719744,    243403776,    21073920,    1161216,     36864,     512;
  8:   702475590656,   944504995840,   569232921600,  202393927680,   46984304640,   7434878976,   811345920,   60211200,   2903040,   81920,   1024;
  9: 10413167579136, 15454462994432, 10389554954240, 4174374758400, 1113166602240, 206730940416, 27261222912, 2549944320, 165580800, 7096320, 180224, 2048;

The row polynomials satisfy (n+8+x)*P(n,x) = (14*n+126+(2*n+32)*x+2*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 252 + 64*x + 4*x^2.

P[0, x_] := 252 + 64*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14*n + 126 + (2*n + 32)*x + 2*x^2)*P[n - 1, x])/(n + 8 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 252 + 64*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14*n + 126 + (2*n + 32)*x + 2*x^2)*v[i - 1])/(n + 8 + 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 differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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
        • 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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A027466 → D+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.