A038281 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:         112,           66,            9;
  1:        1088,         1008,          297,           27;
  2:       10240,        13056,         6048,         1188,           81;
  3:       94208,       153600,        97920,        30240,         4455,         243;
  4:      851968,      1695744,      1382400,       587520,       136080,       16038,         729;
  5:     7602176,     17891328,     17805312,      9676800,      3084480,      571536,       56133,       2187;
  6:    67108864,    182452224,    214695936,    142442496,     58060800,    14805504,     2286144,     192456,       6561;
  7:   587202560,   1811939328,   2463105024,   1932263424,    961486848,   313528320,    66624768,    8817984,     649539,     19683;
  8:  5100273664,  17616076800,  27179089920,  24631050240,  14491975680,  5768921088,  1567641600,  285534720,   33067440,   2165130,   59049;
  9: 44023414784, 168309030912, 290665267200, 298969989120, 203206164480, 95647039488, 31729065984, 7390310400, 1177830720, 121247280, 7144929, 177147;

The row polynomials satisfy (3*n+11+3*x)*P(n,x) = (24*n+112+(9*n+66)*x+9*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 112 + 66*x + 9*x^2.

P[0, x_] := 112 + 66*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24*n + 112 + (9*n + 66)*x + 9*x^2)*P[n - 1, x])/(3*n + 11 + 3*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 112 + 66*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24*n + 112 + (9*n + 66)*x + 9*x^2)*v[i - 1])/(3*n + 11 + 3*x)); 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 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) [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) (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 ?

• A027466 → basis A013610 → D+1

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.