A028326 → basis A013611 → basis A008288

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Submission text ?

Triangle begins:
  0:        2;
  1:       12,         8;
  2:       72,       128,        32;
  3:      432,      1312,       832,        128;
  4:     2592,     11264,     12544,       4608,        512;
  5:    15552,     88192,    145920,      92160,      23552,       2048;
  6:    93312,    653312,   1459712,    1366016,     587776,     114688,      8192;
  7:   559872,   4665856,  13245440,   16828416,   10620928,    3432448,    540672,    32768;
  8:  3359232,  32473088, 112336896,  182894592,  156155904,   73269248,  18874368,  2490368,   131072;
  9: 20155392, 221710336, 906969088, 1816526848, 1988001792, 1254555648, 465043456, 99352576, 11272192, 524288;

The row polynomials satisfy P(n,x) = -(12-8*x)*P(n-2,x) + (8+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 12 + 8*x.

P[0, x_] := 2;
P[1, x_] := 12 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(12 - 8*x)*P[n - 2, x] + (8 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 12 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(12 - 8*x)*v[i - 2] + (8 + 4*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 [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
  by Affine-current basis map theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine-current basis map theorem
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine-current basis map theorem

Proof chains ? JSON

• Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ? step ›

  • from basis_map_A013611 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
        • Geometric basis map (elementary) polytool ? step ›

          • from Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
            • 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
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
                • ↑ Liu–Wang theorem (above)
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • 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
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013609 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).?
            • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A178820 → basis A013611 → basis A008288
• A038208 → basis A013611 → basis A008288
• A013609 → basis A013610 → basis A008288
• A196347 → basis A013611 → basis A008288
• basis-map coefficients of A026729 → basis A013609 → basis A008288
• A013609 → basis A008288
• A085881 → basis A013611 → basis A008288
• basis-map coefficients of A112227 → basis A013609 → basis A008288
• A136572 → basis A013609 → basis A008288
• A085880 → basis A013611 → basis A008288
• A038221 → basis A013611 → basis A008288
• A178821 → basis A013611 → basis A008288
• A178822 → basis A013611 → basis A008288
• basis-map coefficients of A026729, shifted → basis A013609 → basis A008288
• A356546 → basis A013611 → basis A008288
• A013609 → basis A008288 → reverse
• A120909 → basis A013610 → basis A008288
• A156992 → basis A013611 → basis A008288

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.