A013612 → basis A013610 → x ↦ x/(x+1)

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:        36,         252,          441;
  1:       216,        2268,         7938,          9261;
  2:      1296,       18144,        95256,        222264,         194481;
  3:      7776,      136080,       952560,       3333960,        5834430,        4084101;
  4:     46656,      979776,      8573040,      40007520,      105019740,      147027636,        85766121;
  5:    279936,     6858432,     72013536,     420078960,     1470276360,     3087580356,      3602177082,      1801088541;
  6:   1679616,    47029248,    576108288,    4032758016,    17643316320,    49401285696,     86452249968,     86452249968,      37822859361;
  7:  10077696,   317447424,   4444263936,   36294822144,   190547816256,   666917356896,   1556140499424,   2334210749136,    2042434405494,     794280046581;
  8:  60466176,  2116316160,  33331979520,  311098475520,  1905478162560,  8003008282752,  23342107491360,  46684214982720,   61273032164820,   47656802794860,   16679880978201;
  9: 362797056, 13967686656, 244434516480, 2566562423040, 17965936961280, 88033091110272, 308115818885952, 770289547214880, 1348006707626040, 1572674492230380, 1100872144561266, 350277500542221;

The row polynomials satisfy (1+x)*P(n,x) = (6+27*x+21*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 252*x + 441*x^2.

P[0, x_] := 36 + 252*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((6 + 27*x + 21*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 + 252*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6 + 27*x + 21*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_A013610 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 basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.?
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.?
            • ↑ Geometric basis map (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^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
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

• A038255 → basis A013609 → reverse
• A038334 → x ↦ x+1 → x ↦ x+1
• A038328 → basis A013609 → reverse
• A038255 → basis A013609
• A038328 → basis A013609 → D
• A013622 → basis A013611 → x ↦ x/(x+1)
• A038328 → basis A013609
• A038329 → basis A013609 → basis A007318
• A038291 → basis A013611 → basis A007318
• A038255 → basis A013609 → D

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.