A038305 → basis A013610 → reverse

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:          81,          234,           169;
  1:         729,         3159,          4563,           2197;
  2:        6561,        37908,         82134,          79092,          28561;
  3:       59049,       426465,       1232010,        1779570,        1285245,         371293;
  4:      531441,      4605822,      16632135,       32032260,       34701615,       20049822,         4826809;
  5:     4782969,     48361131,     209564901,      504508095,      728733915,      631569393,       304088967,        62748517;
  6:    43046721,    497428776,    2514778812,     7264916568,    13117210470,    15157665432,     10947202812,      4517893224,      815730721;
  7:   387420489,   5036466357,   29099583396,    98076373668,   212498809614,   306942724998,    295574475924,    182974675572,    66074188401,    10604499373;
  8:  3486784401,  50364663570,  327370313205,  1260981947160,  3187482144210,  5524969049964,   6650425708290,   5489240267160,  2973338478045,   954404943570,   137858491849;
  9: 31381059609, 498610169343, 3601073445255, 15604651596105, 45080104610970, 91161989324406, 131678429024142, 135858696612210, 98120169775485, 47243044706715, 13647990693051, 1792160394037;

The row polynomials satisfy P(n,x) = (9+13*x)*P(n-1,x) for n >= 1, with P(0,x) = 81 + 234*x + 169*x^2.

P[0, x_] := 81 + 234*x + 169*x^2;
P[n_, x_] := P[n, x] = Expand[(9 + 13*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 81 + 234*x + 169*x^2; for(n = 1, N, my(i = n + 1); v[i] = (9 + 13*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 (−∞, 0]
  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 Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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 ?

• A038305 → basis A013610
• A038279 → basis A013611 → x ↦ x/(x+1)
• A013610 → basis A013610 → x ↦ x+1
• A038305 → basis A013610 → D

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.