A038336 → D+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:           384,            440,             100;
  1:          6048,          11520,            6600,            1000;
  2:         89856,         241920,          230400,           88000,           10000;
  3:       1285632,        4492800,         6048000,         3840000,         1100000,          100000;
  4:      17915904,       77137920,       134784000,       120960000,        57600000,        13200000,         1000000;
  5:     244850688,     1254113280,      2699827200,      3144960000,      2116800000,       806400000,       154000000,        10000000;
  6:    3296526336,    19588055040,     50164531200,     71995392000,     62899200000,     33868800000,     10752000000,      1760000000,      100000000;
  7:   43858132992,   296687370240,    881462476800,   1504935936000,   1619896320000,   1132185600000,    508032000000,    138240000000,    19800000000,     1000000000;
  8:  577895399424,  4385813299200,  14834368512000,  29382082560000,  37623398400000,  32397926400000,  18869760000000,   7257600000000,  1728000000000,   220000000000,   10000000000;
  9: 7553918435328, 63568493936640, 241219731456000, 543926845440000, 808007270400000, 827714764800000, 593961984000000, 296524800000000, 99792000000000, 21120000000000, 2420000000000, 100000000000;

The row polynomials satisfy (5*n+11+5*x)*P(n,x) = (60*n+192+(50*n+220)*x+50*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 384 + 440*x + 100*x^2.

P[0, x_] := 384 + 440*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((60*n + 192 + (50*n + 220)*x + 50*x^2)*P[n - 1, x])/(5*n + 11 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 384 + 440*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((60*n + 192 + (50*n + 220)*x + 50*x^2)*v[i - 1])/(5*n + 11 + 5*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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.?
    • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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 polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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 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
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Parent-row provers of polytool (above)
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038271 → basis A013609 → D+1
• A038259 → D+1
• A013612 → x ↦ x+1 → 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.