A038328 → basis A013609 → D+1

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Submission text ?

Triangle begins:
  0:            308,            144,             16;
  1:           5096,           3696,            864,             64;
  2:          82320,          81536,          29568,           4608,            256;
  3:        1306144,        1646400,         815360,         197120,          23040,          1024;
  4:       20437312,       31347456,       19756800,        6522880,        1182720,        110592,          4096;
  5:      316240512,      572244736,      438864384,      184396800,       45660160,       6623232,        516096,        16384;
  6:     4849021184,    10119696384,     9155915776,     4681220096,     1475174400,     292225024,      35323904,      2359296,       65536;
  7:    73789452800,   174564762624,   182154534912,   109870989312,    42130980864,   10621255680,    1753350144,    181665792,    10616832,     262144;
  8:  1115696526336,  2951578112000,  3491295252480,  2428727132160,  1098709893120,  337047846912,   70808371200,  10019143680,   908328960,   47185920,   1048576;
  9: 16776769988608, 49090647158784, 64934718464000, 51205663703040, 26715998453760, 9668647059456, 2471684210688, 445081190400, 55105290240, 4440719360, 207618048, 4194304;

The row polynomials satisfy (2*n+9+2*x)*P(n,x) = (28*n+154+(8*n+72)*x+8*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 308 + 144*x + 16*x^2.

P[0, x_] := 308 + 144*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((28*n + 154 + (8*n + 72)*x + 8*x^2)*P[n - 1, x])/(2*n + 9 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 308 + 144*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((28*n + 154 + (8*n + 72)*x + 8*x^2)*v[i - 1])/(2*n + 9 + 2*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 basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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
        • 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

Related ?

Same fingerprint ?

• A038255 → basis A013609 → 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.