A038329 → basis A013611 → reverse

Matches ?

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Submission text ?

Triangle begins:
  0:          144,            360,            225;
  1:         1728,           6480,           8100,            3375;
  2:        20736,         103680,         194400,          162000,           50625;
  3:       248832,        1555200,        3888000,         4860000,         3037500,           759375;
  4:      2985984,       22394880,       69984000,       116640000,       109350000,         54675000,         11390625;
  5:     35831808,      313528320,     1175731200,      2449440000,      3061800000,       2296350000,        956812500,        170859375;
  6:    429981696,     4299816960,    18811699200,     47029248000,     73483200000,      73483200000,      45927000000,      16402500000,      2562890625;
  7:   5159780352,    58047528960,   290237644800,    846526464000,   1587237120000,    1984046400000,    1653372000000,     885735000000,    276792187500,     38443359375;
  8:  61917364224,   773967052800,  4353564672000,  14511882240000,  31744742400000,   47617113600000,   49601160000000,   35429400000000,  16607531250000,   4613203125000,   576650390625;
  9: 743008370688, 10216365096960, 63852281856000, 239446056960000, 598615142400000, 1047576499200000, 1309470624000000, 1169170200000000, 730731375000000, 304471406250000, 76117851562500, 8649755859375;

The row polynomials satisfy P(n,x) = (12+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 360*x + 225*x^2.

P[0, x_] := 144 + 360*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[(12 + 15*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 144 + 360*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 15*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 basis_map_A013611 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

Related ?

Candidate equalities ?

• A038329 → x ↦ x/(x+1) 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A013611 → basis A007318
• A027467 → basis A013611 → x ↦ x/(x+1)
• A038271 → basis A013610
• A038231 → x ↦ x/(x+1)
• A013611 → D → basis A007318
• A013611 → basis A007318 → reverse
• A038271 → basis A013610 → reverse
• A038271 → basis A013611 → basis A013609
• A038319 → basis A013611
• basis-map coefficients of A112227 → basis A013611 → x ↦ x+1
• A038334 → basis A013609
• A130123 → basis A013611 → x ↦ x+1
• A038231 → D → basis A013611
• basis-map coefficients of A026729 → basis A013611 → x ↦ x+1
• A120910 → basis A013609 → basis A013609
• A038329 → basis A013611 → D
• A038334 → basis A013609 → D
• A038271 → basis A013610 → D
• A038319 → basis A013611 → D
• A038329 → basis A013611
• A013622 → basis A013609 → D
• A136572 → basis A013611 → basis A007318
• A013622 → 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.