A139548 → D+1

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Submission text ?

Triangle begins:
  0:    1;
  1:    6,      8,       2;
  2:   20,     64,      72,      32,        4;
  3:   56,    288,     600,     640,      360,       96,         8;
  4:  144,   1024,    3136,    5376,     5600,     3584,      1344,       256,        16;
  5:  352,   3200,   12960,   30720,    47040,    48384,     33600,     15360,      4320,       640,        32;
  6:  832,   9216,   46464,  140800,   285120,   405504,    413952,    304128,    158400,     56320,     12672,      1536,       64;
  7: 1920,  25088,  151424,  559104,  1409408,  2562560,   3459456,   3514368,   2690688,   1537536,    640640,    186368,    34944,     3584,     128;
  8: 4352,  65536,  460800, 2007040,  6056960, 13418496,  22550528,  29286400,  29652480,  23429120,  14350336,   6709248,  2329600,   573440,   92160,    8192,    256;
  9: 9728, 165888, 1331712, 6684672, 23500800, 61415424, 123561984, 195526656, 246445056, 248934400, 201636864, 130351104, 66533376, 26320896, 7833600, 1671168, 235008, 18432, 512;

The row polynomials satisfy (2*n-1+x)*P(n,x) = (4*n+2+(8*n+6)*x+(4*n+6)*x^2+2*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 2 + (8*n + 6)*x + (4*n + 6)*x^2 + 2*x^3)*P[n - 1, x])/(2*n - 1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 2 + (8*n + 6)*x + (4*n + 6)*x^2 + 2*x^3)*v[i - 1])/(2*n - 1 + 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

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] 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)

Related ?

Same fingerprint ?

• A034870 → D+1
• A034870 → basis A013609 → θ+1
• A139548 → basis A013609 → θ+1
• A034870 → basis A013609 → D+1
• A139548 → θ+1
• A139548 → basis A013609 → D+1
• A034870 → θ+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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.