A038334 → x ↦ x+1 → D

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Submission text ?

Triangle begins:
  0:             320,              128;
  1:            9600,             7680,             1536;
  2:          256000,           307200,           122880,            16384;
  3:         6400000,         10240000,          6144000,          1638400,           163840;
  4:       153600000,        307200000,        245760000,         98304000,         19660800,          1572864;
  5:      3584000000,       8601600000,       8601600000,       4587520000,       1376256000,        220200960,        14680064;
  6:     81920000000,     229376000000,     275251200000,     183500800000,      73400320000,      17616076800,      2348810240,       134217728;
  7:   1843200000000,    5898240000000,    8257536000000,    6606028800000,    3303014400000,    1056964608000,    211392921600,     24159191040,     1207959552;
  8:  40960000000000,  147456000000000,  235929600000000,  220200960000000,  132120576000000,   52848230400000,  14092861440000,   2415919104000,   241591910400,   10737418240;
  9: 901120000000000, 3604480000000000, 6488064000000000, 6920601600000000, 4844421120000000, 2325322137600000, 775107379200000, 177167400960000, 26575110144000, 2362232012800, 94489280512;

The row polynomials satisfy (n+1)*P(n,x) = (20*n+40+(8*n+16)*x)*P(n-1,x) for n >= 1, with P(0,x) = 320 + 128*x.

P[0, x_] := 320 + 128*x;
P[n_, x_] := P[n, x] = Expand[Together[((20*n + 40 + (8*n + 16)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 320 + 128*x; for(n = 1, N, my(i = n + 1); v[i] = ((20*n + 40 + (8*n + 16)*x)*v[i - 1])/(n + 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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 → reverse
• A038220 → basis A007318
• A038291 → basis A013611 → D
• A038327 → basis A013609 → basis A013610
• A013620 → x ↦ x/(x+1)
• A013622 → basis A013611 → reverse
• A038291 → basis A013611 → reverse
• A013622 → basis A013611
• A038221 → basis A013610 → x ↦ x/(x+1)
• A028326 → basis A013610 → x ↦ x/(x+1)
• A085880 → basis A013610 → x ↦ x/(x+1)
• A196347 → basis A013610 → x ↦ x/(x+1)
• A120910 → basis A013609 → basis A007318
• A038329 → basis A013609
• A038291 → basis A013611
• A178821 → basis A013610 → x ↦ x/(x+1)
• A038334 → x ↦ x+1
• A038231 → D → basis A013609
• A156992 → basis A013610 → x ↦ x/(x+1)
• A038329 → basis A013609 → D
• A038220 → basis A007318 → D
• A038208 → basis A013610 → x ↦ x/(x+1)
• A178820 → basis A013610 → x ↦ x/(x+1)
• A038338 → basis A013610 → x ↦ x/(x+1)
• A038334 → x ↦ x+1 → reverse
• A013622 → basis A013611 → D
• A178822 → basis A013610 → x ↦ x/(x+1)
• A013612 → basis A013610
• A013612 → 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.