A038226 → D

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Submission text ?

Triangle begins:
  0:      48,       128;
  1:     216,      1152,       1536;
  2:     864,      6912,      18432,       16384;
  3:    3240,     34560,     138240,      245760,      163840;
  4:   11664,    155520,     829440,     2211840,     2949120,      1572864;
  5:   40824,    653184,    4354560,    15482880,    30965760,     33030144,     14680064;
  6:  139968,   2612736,   20901888,    92897280,   247726080,    396361728,    352321536,    134217728;
  7:  472392,  10077696,   94058496,   501645312,  1672151040,   3567255552,   4756340736,   3623878656,   1207959552;
  8: 1574640,  37791360,  403107840,  2508226560, 10032906240,  26754416640,  47563407360,  54358179840,  36238786560,  10737418240;
  9: 5196312, 138568320, 1662819840, 11824496640, 55180984320, 176579149824, 392398110720, 597939978240, 597939978240, 354334801920, 94489280512;

The row polynomials satisfy (n+1)*P(n,x) = (3*n+6+(8*n+16)*x)*P(n-1,x) for n >= 1, with P(0,x) = 48 + 128*x.

P[0, x_] := 48 + 128*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 6 + (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] = 48 + 128*x; for(n = 1, N, my(i = n + 1); v[i] = ((3*n + 6 + (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 Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • 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
        • 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
    • ↑ Iterated real-rootedness preservers (above)
    • ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

• A178821 → basis A013611 → basis A013611
• A038329 → basis A013610 → basis A007318
• A013622 → x ↦ x/(x+1) → D
• A178820 → basis A013611 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013611
• A027466 → basis A013610 → D
• A136572 → basis A013609 → basis A013611
• A038281 → D
• A120909 → basis A013611
• basis-map coefficients of A026729 → basis A013609 → basis A013611
• A038208 → basis A013611 → basis A013611
• A038328 → x ↦ x+1 → basis A013610
• A013609 → basis A013610 → basis A013611
• Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• basis-map coefficients of A112227 → basis A013609 → basis A013611
• A038271 → basis A013611 → x ↦ x/(x+1)
• A013622 → x ↦ x/(x+1)
• A028326 → basis A013611 → basis A013611
• A027466 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013611
• A013612 → basis A013609 → x ↦ x/(x+1)
• A038327 → basis A013610 → basis A013609
• A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
• A120909 → basis A013610 → basis A013611
• A013609 → D → basis A013611
• A130123 → basis A013609 → 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.