A038281 → D

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Submission text ?

Triangle begins:
  0:          48,           18;
  1:         576,          432,           81;
  2:        6144,         6912,         2592,          324;
  3:       61440,        92160,        51840,        12960,         1215;
  4:      589824,      1105920,       829440,       311040,        58320,        4374;
  5:     5505024,     12386304,     11612160,      5806080,      1632960,      244944,       15309;
  6:    50331648,    132120576,    148635648,     92897280,     34836480,     7838208,      979776,      52488;
  7:   452984832,   1358954496,   1783627776,   1337720832,    627056640,   188116992,    35271936,    3779136,    177147;
  8:  4026531840,  13589544960,  20384317440,  17836277760,  10032906240,  3762339840,   940584960,  151165440,  14171760,   590490;
  9: 35433480192, 132875550720, 224227491840, 224227491840, 147149291520, 66217181184, 20692869120, 4434186240, 623557440, 51963120, 1948617;

The row polynomials satisfy (n+1)*P(n,x) = (8*n+16+(3*n+6)*x)*P(n-1,x) for n >= 1, with P(0,x) = 48 + 18*x.

P[0, x_] := 48 + 18*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 16 + (3*n + 6)*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 + 18*x; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 16 + (3*n + 6)*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 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 ?

• A028326 → basis A013611 → basis A013611
• A013609 → basis A013610 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013611
• A038226 → D
• A178821 → basis A013611 → basis A013611
• A120909 → basis A013611
• A013622 → x ↦ x/(x+1) → D
• A013609 → D → basis A013611
• basis-map coefficients of A026729 → basis A013609 → basis A013611
• A013622 → x ↦ x/(x+1)
• A013612 → basis A013609 → x ↦ x/(x+1)
• A027466 → basis A013610
• A178820 → basis A013611 → basis A013611
• A136572 → basis A013609 → basis A013611
• A027466 → basis A013610 → D
• A120909 → basis A013610 → basis A013611
• A038329 → basis A013610 → basis A007318
• A038328 → x ↦ x+1 → basis A013610
• A130123 → basis A013609 → basis A013611
• A038208 → basis A013611 → basis A013611
• basis-map coefficients of A112227 → basis A013609 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013611
• A038327 → basis A013610 → basis A013609
• A038271 → basis A013611 → x ↦ x/(x+1)
• Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.

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.