A038331 → x ↦ x+1 → x ↦ x/(x+1)

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Submission text ?

Triangle begins:
  0:            289,             748,              484;
  1:           4913,           19074,            24684,             10648;
  2:          83521,          432344,           839256,            724064,            234256;
  3:        1419857,         9187310,         23778920,          30772720,          19911760,           5153632;
  4:       24137569,       187421124,        606362460,        1046272480,        1015499760,         525670464,         113379904;
  5:      410338673,      3717185626,      14431426548,       31126606280,       40281490480,       31277392608,       13492208576,        2494357888;
  6:     6975757441,     72219606448,     327112335088,      846643690816,     1369570676320,     1417908464896,      917470183168,      339232672768,       54875873536;
  7:   118587876497,   1381199973318,    7149741038352,    21589414115808,    41908862695392,    54234998782272,    46790979341568,    25951299466752,     8396008651008,     1207269217792;
  8:  2015993900449,  26089332829340,  151931997064980,   524314342812480,  1187417776369440,  1843989958597248,  1988616622016640,  1470573636449280,   713660735335680,   205235767024640,   26559922791424;
  9: 34271896307633, 487870523908658, 3156809272350140, 12255847763241720, 31721017740155040, 57471020376280896, 74374261663422336, 68749317504003840, 44484852502590720, 19189544216803840, 4966705561996288, 584318301411328;

The row polynomials satisfy (1+x)*P(n,x) = (17+39*x+22*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 748*x + 484*x^2.

P[0, x_] := 289 + 748*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((17 + 39*x + 22*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 289 + 748*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((17 + 39*x + 22*x^2)*v[i - 1])/(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 [-1, 0]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^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)*5^j.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Same fingerprint ?

• A038336 → x ↦ x/(x+1) → basis A013609

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.