A038329 → basis A013609 → x ↦ x/(x+1)

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Submission text ?

Triangle begins:
  0:           225,             630,             441;
  1:          3375,           14175,           19845,             9261;
  2:         50625,          283500,          595350,           555660,            194481;
  3:        759375,         5315625,        14883750,         20837250,          14586075,           4084101;
  4:      11390625,        95681250,       334884375,        625117500,         656373375,         367569090,          85766121;
  5:     170859375,      1674421875,      7032571875,      16409334375,       22973068125,       19297377225,        9005442705,        1801088541;
  6:    2562890625,     28704375000,    140651437500,     393824025000,      689192043750,      771895089000,      540326562300,      216130624920,       37822859361;
  7:   38443359375,    484386328125,   2712563437500,    8861040562500,    18608185181250,    26051459253750,    24314695303500,    14588817182100,     5106086013735,     794280046581;
  8:  576650390625,   8073105468750,  50860564453125,  189879440625000,   465204629531250,   781543777612500,   911801073881250,   729440859105000,   382956451030125,  119142006987150,   16679880978201;
  9: 8649755859375, 133206240234375, 932443681640625, 3916263462890625, 10965537696093750, 21492453884343750, 30089435438081250, 30089435438081250, 21062604806656875, 9829215576439875, 2752180361403165, 350277500542221;

The row polynomials satisfy (1+x)*P(n,x) = (15+36*x+21*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 225 + 630*x + 441*x^2.

P[0, x_] := 225 + 630*x + 441*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((15 + 36*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] = 225 + 630*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((15 + 36*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_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) 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.?
            • 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
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A038271: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.
• A038220 → basis A007318 → x ↦ x/(x+1)
• A038271 → D
• A013612 → basis A013610 → basis A007318
• A013622 → basis A013611 → basis A007318
• A038271 → reverse
• A038334 → x ↦ x+1 → x ↦ x/(x+1)
• A038291 → basis A013611 → x ↦ x/(x+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.