A038329 → reverse

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Submission text ?

Triangle begins:
  0:      9,      72,       144;
  1:     27,     324,      1296,       1728;
  2:     81,    1296,      7776,      20736,       20736;
  3:    243,    4860,     38880,     155520,      311040,      248832;
  4:    729,   17496,    174960,     933120,     2799360,     4478976,      2985984;
  5:   2187,   61236,    734832,    4898880,    19595520,    47029248,     62705664,     35831808;
  6:   6561,  209952,   2939328,   23514624,   117573120,   376233984,    752467968,    859963392,     429981696;
  7:  19683,  708588,  11337408,  105815808,   634894848,  2539579392,   6772211712,  11609505792,   11609505792,    5159780352;
  8:  59049, 2361960,  42515280,  453496320,  3174474240, 15237476352,  50791587840, 116095057920,  174142586880,  154793410560,   61917364224;
  9: 177147, 7794468, 155889360, 1870672320, 14965378560, 83806119936, 335224479744, 957784227840, 1915568455680, 2554091274240, 2043273019392, 743008370688;

The row polynomials satisfy P(n,x) = (3+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 72*x + 144*x^2.

P[0, x_] := 9 + 72*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(3 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 9 + 72*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 12*x)*v[i - 1]); 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 (−∞, 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 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 ?

• basis-map coefficients of A030528, shifted → (θ+1)² → basis A013611
• A132440 → θ+1 → basis A013611
• A027466 → basis A013609
• A130123 → basis A013611 → D
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A013611
• basis-map coefficients of A026729 → D → basis A013611
• A120910 → basis A007318 → basis A007318
• A130123 → basis A013611
• basis-map coefficients of A026729 → (θ+1)² → basis A013611
• A136521 → basis A013611
• basis-map coefficients of A112227 → basis A013611 → D
• basis-map coefficients of A030528, shifted → D → basis A013611
• A038329 → D
• A027467 → basis A013611 → reverse
• A013611 → D → D
• image under θ+1 → (θ+1)² → basis A013611
• A027467 → basis A013611 → D
• A038330 → basis A007318 → reverse
• A127648 → D → basis A013611
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• A136572 → basis A013611 → reverse
• basis-map coefficients of A021012 → θ+1 → basis A013611
• A038231 → D
• A130123 → D → basis A013611
• A136572 → basis A013611
• basis-map coefficients of A112227 → (θ+1)² → basis A013611
• A132440 → basis A013611
• A038330 → basis A007318 → D
• basis-map coefficients of A030528, shifted → basis A013611
• A131431 → θ+1 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013611
• A136572 → (θ+1)² → basis A013611
• A136572 → basis A013611 → D
• A130123 → θ+1 → basis A013611
• A013611 → D
• A038303 → basis A007318 → basis A013610
• A131431 → basis A013611
• A136572 → D → basis A013611
• basis-map coefficients of A026729 → basis A013611
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• basis-map coefficients of A112227 → basis A013611
• A038231 → D → D
• basis-map coefficients of A026729, shifted → θ+1 → basis A013611
• A131431 → D → basis A013611
• A130123 → (θ+1)² → basis A013611
• A129186 → basis A013611
• A131436 → basis A013611
• basis-map coefficients of A026729 → basis A013611 → D
• basis-map coefficients of A026729, shifted → D → basis A013611
• basis-map coefficients of A030528, shifted → θ+1 → basis A013611
• image under θ+1 → θ+1 → D → basis A013611
• A038330 → basis A007318
• basis-map coefficients of A021012 → basis A013611
• A027466 → basis A013609 → D
• A027467 → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• A013610 → basis A013611 → x ↦ x/(x+1)
• A134309 → basis A013611
• basis-map coefficients of A112227 → basis A013611 → reverse
• image under θ+1 → basis A013611
• A136572 → θ+1 → basis A013611
• basis-map coefficients of A026729 → basis A013611 → reverse
• basis-map coefficients of A112227 → θ+1 → basis A013611

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.