A038329 → D

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Submission text ?

Triangle begins:
  0:            72,            18;
  1:          1296,           648,            81;
  2:         20736,         15552,          3888,           324;
  3:        311040,        311040,        116640,         19440,          1215;
  4:       4478976,       5598720,       2799360,        699840,         87480,         4374;
  5:      62705664,      94058496,      58786560,      19595520,       3674160,       367416,        15309;
  6:     859963392,    1504935936,    1128701952,     470292480,     117573120,     17635968,      1469664,       52488;
  7:   11609505792,   23219011584,   20316635136,   10158317568,    3174474240,    634894848,     79361856,     5668704,     177147;
  8:  154793410560,  348285173760,  348285173760,  203166351360,   76187381760,  19046845440,   3174474240,   340122240,   21257640,   590490;
  9: 2043273019392, 5108182548480, 5746705367040, 3831136911360, 1676122398720, 502836719616, 104757649920, 14965378560, 1403004240, 77944680, 1948617;

The row polynomials satisfy (n+1)*P(n,x) = (12*n+24+(3*n+6)*x)*P(n-1,x) for n >= 1, with P(0,x) = 72 + 18*x.

P[0, x_] := 72 + 18*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 24 + (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] = 72 + 18*x; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 24 + (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)*12^(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
    • 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

Related ?

Same fingerprint ?

• basis-map coefficients of A021012 → basis A013611
• basis-map coefficients of A026729 → basis A013611
• A027467 → basis A013611
• A132440 → θ+1 → basis A013611
• A027466 → basis A013609
• basis-map coefficients of A030528, shifted → D → basis A013611
• A013611 → D → D
• A136521 → basis A013611
• A134309 → basis A013611
• A027467 → basis A013611 → D
• A038329 → reverse
• basis-map coefficients of A030528, shifted → θ+1 → basis A013611
• A136572 → (θ+1)² → basis A013611
• A136572 → D → basis A013611
• A131436 → basis A013611
• A027466 → basis A013609 → D
• image under θ+1 → (θ+1)² → basis A013611
• A038303 → basis A007318 → basis A013610
• basis-map coefficients of A112227 → basis A013611
• A131431 → basis A013611
• basis-map coefficients of A026729, shifted → θ+1 → basis A013611
• basis-map coefficients of A112227 → basis A013611 → D
• A038330 → basis A007318 → reverse
• basis-map coefficients of A026729 → (θ+1)² → basis A013611
• A038231 → D
• A136572 → basis A013611 → reverse
• A130123 → basis A013611 → D
• A038330 → basis A007318 → D
• A130123 → θ+1 → basis A013611
• A130123 → basis A013611
• A131431 → D → basis A013611
• A131431 → θ+1 → basis A013611
• A027467 → basis A013611 → reverse
• A130123 → D → basis A013611
• A129186 → basis A013611
• A132440 → basis A013611
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• basis-map coefficients of A112227 → basis A013611 → reverse
• basis-map coefficients of A026729 → D → basis A013611
• A130123 → (θ+1)² → basis A013611
• A136572 → basis A013611
• basis-map coefficients of A026729, shifted → D → basis A013611
• basis-map coefficients of A026729, shifted → basis A013611
• A013611 → D
• basis-map coefficients of A112227 → θ+1 → basis A013611
• A136572 → basis A013611 → D
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• A038330 → basis A007318
• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013611
• A127648 → D → basis A013611
• A038231 → D → D
• basis-map coefficients of A030528, shifted → basis A013611
• basis-map coefficients of A021012 → θ+1 → basis A013611
• basis-map coefficients of A026729 → basis A013611 → D
• image under θ+1 → θ+1 → D → basis A013611
• basis-map coefficients of A026729 → basis A013611 → reverse
• A120910 → basis A007318 → basis A007318
• basis-map coefficients of A112227 → (θ+1)² → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• image under θ+1 → basis A013611
• A013610 → basis A013611 → x ↦ x/(x+1)
• A136572 → θ+1 → basis A013611
• basis-map coefficients of A026729, shifted → (θ+1)² → 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.