A038334 → x ↦ x+1 → reverse

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Submission text ?

Triangle begins:
  0:         64,          320,           400;
  1:        512,         3840,          9600,           8000;
  2:       4096,        40960,        153600,         256000,          160000;
  3:      32768,       409600,       2048000,        5120000,         6400000,         3200000;
  4:     262144,      3932160,      24576000,       81920000,       153600000,       153600000,        64000000;
  5:    2097152,     36700160,     275251200,     1146880000,      2867200000,      4300800000,      3584000000,       1280000000;
  6:   16777216,    335544320,    2936012800,    14680064000,     45875200000,     91750400000,    114688000000,      81920000000,      25600000000;
  7:  134217728,   3019898880,   30198988800,   176160768000,    660602880000,   1651507200000,   2752512000000,    2949120000000,    1843200000000,     512000000000;
  8: 1073741824,  26843545600,  301989888000,  2013265920000,   8808038400000,  26424115200000,  55050240000000,   78643200000000,   73728000000000,   40960000000000,  10240000000000;
  9: 8589934592, 236223201280, 2952790016000, 22145925120000, 110729625600000, 387553689600000, 968884224000000, 1730150400000000, 2162688000000000, 1802240000000000, 901120000000000, 204800000000000;

The row polynomials satisfy P(n,x) = (8+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 64 + 320*x + 400*x^2.

P[0, x_] := 64 + 320*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 20*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 64 + 320*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (8 + 20*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)*8^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)*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)
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A038220 → basis A007318 → D
• A013620 → x ↦ x/(x+1)
• A178821 → basis A013610 → x ↦ x/(x+1)
• A038208 → basis A013610 → x ↦ x/(x+1)
• A038334 → x ↦ x+1
• A038220 → basis A007318
• A038291 → basis A013611 → reverse
• A038291 → basis A013611
• A196347 → basis A013610 → x ↦ x/(x+1)
• A038291 → basis A013611 → D
• A156992 → basis A013610 → x ↦ x/(x+1)
• A013612 → basis A013610
• A028326 → basis A013610 → x ↦ x/(x+1)
• A013622 → basis A013611
• A038221 → basis A013610 → x ↦ x/(x+1)
• A038220 → basis A007318 → reverse
• A013612 → basis A013610 → D
• A178820 → basis A013610 → x ↦ x/(x+1)
• A178822 → basis A013610 → x ↦ x/(x+1)
• A038231 → D → basis A013609
• A038327 → basis A013609 → basis A013610
• A038329 → basis A013609 → D
• A120910 → basis A013609 → basis A007318
• A038334 → x ↦ x+1 → D
• A038338 → basis A013610 → x ↦ x/(x+1)
• A013622 → basis A013611 → D
• A085880 → basis A013610 → x ↦ x/(x+1)
• A013622 → basis A013611 → reverse
• A038329 → basis A013609

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.