A139548 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:          0;
  1:         24,          36;
  2:        384,        1728,          2592,          1296;
  3:       4608,       34560,        103680,        155520,         116640,           34992;
  4:      49152,      516096,       2322432,       5806080,        8709120,         7838208,         3919104,           839808;
  5:     491520,     6635520,      39813120,     139345920,      313528320,       470292480,       470292480,        302330880,        113374080,         18895680;
  6:    4718592,    77856768,     583925760,    2627665920,     7882997760,     16554295296,     24831442944,      26605117440,      19953838080,       9976919040,       2993075712,        408146688;
  7:   44040192,   858783744,    7729053696,   42509795328,   159411732480,    430411677696,    860823355392,    1291235033088,    1452639412224,    1210532843520,     726319706112,     297130788864,      74282697216,       8571080448;
  8:  402653184,  9059696640,   95126814720,  618324295680,  2782459330560,   9182115790848,  22955289477120,   44270915420160,   66406373130240,   77474101985280,   69726691786752,   47540926218240,   23770463109120,    8228237230080,   1763193692160,    176319369216;
  9: 3623878656, 92408905728, 1108906868736, 8316801515520, 43663207956480, 170286511030272, 510859533090816, 1204168899428352, 2257816686428160, 3386725029642240, 4064070035570688, 3879339579408384, 2909504684556288, 1678560394936320, 719383026401280, 215814907920384, 40465295235072, 3570467226624;

The row polynomials satisfy (n-1)*P(n,x) = (8*n+24*n*x+18*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 24 + 36*x.

P[0, x_] := 0;
P[1, x_] := 24 + 36*x;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + 24*n*x + 18*n*x^2)*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] = 0; v[2] = 24 + 36*x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n + 24*n*x + 18*n*x^2)*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

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013610 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
            • 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
            • 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 (above)
        • ↑ Iterated real-rootedness preservers (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A007318 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, …?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.?
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)

Related ?

Same fingerprint ?

• A034870 → D → basis A013610
• A034870 → 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.