A013612 → x ↦ x+1 → x ↦ x/(x+1)

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:        36,        132,         121;
  1:       216,       1188,        2178,         1331;
  2:      1296,       9504,       26136,        31944,         14641;
  3:      7776,      71280,      261360,       479160,        439230,        161051;
  4:     46656,     513216,     2352240,      5749920,       7906140,       5797836,       1771561;
  5:    279936,    3592512,    19758816,     60374160,     110685960,     121754556,      74405562,      19487171;
  6:   1679616,   24634368,   158070528,    579591936,    1328231520,    1948072896,    1785733488,     935384208,     214358881;
  7:  10077696,  166281984,  1219401216,   5216327424,   14344900416,   26298984096,   32143202784,   25255373616,   11575379574,    2357947691;
  8:  60466176, 1108546560,  9145509120,  44711377920,  143449004160,  315587809152,  482148041760,  505107472320,  347261387220,  141476861460,   25937424601;
  9: 362797056, 7316407296, 67067066880, 368868867840, 1352519182080, 3471465900672, 6364354151232, 8334273293280, 7639750518840, 4668736428180, 1711870023666, 285311670611;

The row polynomials satisfy (1+x)*P(n,x) = (6+17*x+11*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 132*x + 121*x^2.

P[0, x_] := 36 + 132*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6 + 17*x + 11*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] = 36 + 132*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6 + 17*x + 11*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 of coefficients in expansion of (1+5x)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^n.?
        • 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 of coefficients in expansion of (1+5x)^n.?
        • ↑ Liu–Wang theorem (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)

Related ?

Candidate equalities ?

• A038320 → reverse 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A038271 → basis A013609 → x ↦ x/(x+1)
• A038334 → basis A013610 → basis A007318
• A038259 → x ↦ x/(x+1)
• A038277 → basis A013610
• A038320 → D
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
• A038314 → x ↦ x+1
• A038336 → x ↦ x/(x+1) → D
• A038336 → x ↦ x/(x+1)
• A038314 → x ↦ x+1 → D
• A027466 → basis A013610 → 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.