A038330 → basis A007318 → reverse

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Submission text ?

Triangle begins:
  0:      16,       128,        256;
  1:      64,       768,       3072,        4096;
  2:     256,      4096,      24576,       65536,        65536;
  3:    1024,     20480,     163840,      655360,      1310720,       1048576;
  4:    4096,     98304,     983040,     5242880,     15728640,      25165824,      16777216;
  5:   16384,    458752,    5505024,    36700160,    146800640,     352321536,     469762048,      268435456;
  6:   65536,   2097152,   29360128,   234881024,   1174405120,    3758096384,    7516192768,     8589934592,     4294967296;
  7:  262144,   9437184,  150994944,  1409286144,   8455716864,   33822867456,   90194313216,   154618822656,   154618822656,    68719476736;
  8: 1048576,  41943040,  754974720,  8053063680,  56371445760,  270582939648,  901943132160,  2061584302080,  3092376453120,  2748779069440,  1099511627776;
  9: 4194304, 184549376, 3690987520, 44291850240, 354334801920, 1984274890752, 7937099563008, 22677427322880, 45354854645760, 60473139527680, 48378511622144, 17592186044416;

The row polynomials satisfy P(n,x) = (4+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 16 + 128*x + 256*x^2.

P[0, x_] := 16 + 128*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(4 + 16*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 16 + 128*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (4 + 16*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_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • Geometric basis map (elementary) 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)
    • 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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038330 → basis A007318
• A027467 → basis A013611
• A136572 → basis A013611 → D
• image under θ+1 → θ+1 → D → basis A013611
• A134309 → basis A013611
• A132440 → θ+1 → basis A013611
• image under θ+1 → (θ+1)² → basis A013611
• basis-map coefficients of A030528, shifted → θ+1 → basis A013611
• A013611 → D
• A130123 → basis A013611 → D
• A136572 → basis A013611
• basis-map coefficients of A026729 → basis A013611
• A129186 → basis A013611
• A130123 → θ+1 → basis A013611
• basis-map coefficients of A112227 → θ+1 → basis A013611
• A027466 → basis A013609
• A136572 → D → basis A013611
• A038231 → D
• basis-map coefficients of A026729 → basis A013611 → reverse
• A131431 → basis A013611
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• basis-map coefficients of A021012 → basis A013611
• A131431 → θ+1 → basis A013611
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013611
• basis-map coefficients of A026729 → basis A013611 → D
• A038329 → reverse
• A027467 → basis A013611 → reverse
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013611
• basis-map coefficients of A112227 → basis A013611 → reverse
• A013611 → D → D
• A130123 → (θ+1)² → basis A013611
• A027466 → basis A013609 → D
• basis-map coefficients of A112227 → (θ+1)² → basis A013611
• basis-map coefficients of A030528, shifted → D → basis A013611
• A013610 → basis A013611 → x ↦ x/(x+1)
• A038329 → D
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• A136572 → θ+1 → basis A013611
• A136521 → basis A013611
• basis-map coefficients of A026729 → D → basis A013611
• A038330 → basis A007318 → D
• A132440 → basis A013611
• A130123 → D → basis A013611
• basis-map coefficients of A030528, shifted → basis A013611
• basis-map coefficients of A021012 → θ+1 → basis A013611
• A120910 → basis A007318 → basis A007318
• A038231 → D → D
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• basis-map coefficients of A112227 → basis A013611
• A136572 → (θ+1)² → basis A013611
• A136572 → basis A013611 → reverse
• A131431 → D → basis A013611
• image under θ+1 → basis A013611
• A027467 → basis A013611 → D
• basis-map coefficients of A026729 → (θ+1)² → basis A013611
• A131436 → basis A013611
• basis-map coefficients of A026729, shifted → θ+1 → basis A013611
• A038303 → basis A007318 → basis A013610
• basis-map coefficients of A112227 → basis A013611 → D
• basis-map coefficients of A026729, shifted → D → basis A013611
• A130123 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013611
• A127648 → D → 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.