A038329 → basis A013610 → basis A007318

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:              576,              432,                81;
  1:            13824,            15552,              5832,               729;
  2:           331776,           497664,            279936,             69984,             6561;
  3:          7962624,         14929920,          11197440,           4199040,           787320,            59049;
  4:        191102976,        429981696,         403107840,         201553920,         56687040,          8503056,           531441;
  5:       4586471424,      12039487488,       13544423424,        8465264640,       3174474240,        714256704,         89282088,         4782969;
  6:     110075314176,     330225942528,      433421549568,      325066162176,     152374763520,      45712429056,       8571080448,       918330048,       43046721;
  7:    2641807540224,    8916100448256,    13374150672384,    11702381838336,    6582589784064,    2468471169024,     617117792256,     99179645184,     9298091736,      387420489;
  8:   63403380965376,  237762678620160,   401224520171520,   401224520171520,  263303591362560,  118486616113152,   37027067535360,   7934371614720,  1115771008320,    92980917360,   3486784401;
  9: 1521681143169024, 6276934715572224, 11769252591697920, 13240409165660160, 9930306874245120, 5213411108978688, 1955029165867008, 523668526571520, 98187848732160, 12273481091520, 920511081864, 31381059609;

The row polynomials satisfy P(n,x) = (24+9*x)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 432*x + 81*x^2.

P[0, x_] := 576 + 432*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[(24 + 9*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 576 + 432*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = (24 + 9*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)*3^j.?
    • Geometric basis map (elementary) polytool ? step ›

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

Related ?

Same fingerprint ?

• A013622 → x ↦ x/(x+1)
• basis-map coefficients of A112227 → basis A013609 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013611
• A038226 → D
• A178820 → basis A013611 → basis A013611
• A038327 → basis A013610 → basis A013609
• A120909 → basis A013610 → basis A013611
• A130123 → basis A013609 → basis A013611
• A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
• A038271 → basis A013611 → x ↦ x/(x+1)
• basis-map coefficients of A026729 → basis A013609 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013611
• A028326 → basis A013611 → basis A013611
• A038281 → D
• A136572 → basis A013609 → basis A013611
• A027466 → basis A013610 → D
• Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• A013609 → D → basis A013611
• A178821 → basis A013611 → basis A013611
• A038208 → basis A013611 → basis A013611
• A027466 → basis A013610
• A038328 → x ↦ x+1 → basis A013610
• A013622 → x ↦ x/(x+1) → D
• A120909 → basis A013611
• A013612 → basis A013609 → x ↦ x/(x+1)
• A013609 → basis A013610 → basis A013611

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.