A038271 → basis A013610 → x ↦ x/(x+1)

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Submission text ?

Triangle begins:
  0:          1;
  1:         12,           27;
  2:        144,          648,          729;
  3:       1728,        11664,        26244,         19683;
  4:      20736,       186624,       629856,        944784,         531441;
  5:     248832,      2799360,     12597120,      28343520,       31886460,       14348907;
  6:    2985984,     40310784,    226748160,     680244480,     1147912560,     1033121304,      387420489;
  7:   35831808,    564350976,   3809369088,   14285134080,    32141551680,    43391094768,    32543321076,    10460353203;
  8:  429981696,   7739670528,  60949905408,  274274574336,   771397240320,  1388515032576,  1562079411648,  1004193907488,   282429536481;
  9: 5159780352, 104485552128, 940369969152, 4936942338048, 16662180390912, 37489905879552, 56234858819328, 54226471004352, 30502389939948, 7625597484987;

The row polynomials satisfy (1+x)*P(n,x) = (12+39*x+27*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((12 + 39*x + 27*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((12 + 39*x + 27*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 whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^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 ?

• A013610 → D → basis A013610
• A013622 → basis A013609 → x ↦ x/(x+1)
• A013610 → basis A013610 → D
• A013610 → basis A013611 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013610
• basis-map coefficients of A026729 → basis A013610 → basis A013610
• A038334 → basis A013609 → basis A007318
• A038329 → basis A013611 → x ↦ x+1
• A038328 → basis A013609 → basis A013609
• basis-map coefficients of A112227 → basis A013610 → basis A013610
• A038279 → basis A013611 → D
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013610
• A013610 → basis A013610
• A136572 → basis A013610 → basis A013610
• A038279 → basis A013611
• A038319 → basis A013611 → x ↦ x/(x+1)
• A130123 → basis A013610 → basis A013610

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.