A038329 → basis A013611 → x ↦ x+1

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Submission text ?

Triangle begins:
  0:              729,               648,               144;
  1:            19683,             26244,             11664,              1728;
  2:           531441,            944784,            629856,            186624,             20736;
  3:         14348907,          31886460,          28343520,          12597120,           2799360,            248832;
  4:        387420489,        1033121304,        1147912560,         680244480,         226748160,          40310784,           2985984;
  5:      10460353203,       32543321076,       43391094768,       32141551680,       14285134080,        3809369088,         564350976,         35831808;
  6:     282429536481,     1004193907488,     1562079411648,     1388515032576,      771397240320,      274274574336,       60949905408,       7739670528,        429981696;
  7:    7625597484987,    30502389939948,    54226471004352,    56234858819328,    37489905879552,    16662180390912,     4936942338048,     940369969152,     104485552128,      5159780352;
  8:  205891132094649,   915071698198440,  1830143396396880,  2169058840174080,  1687045764579840,   899757741109248,   333243607818240,   84633297223680,   14105549537280,   1393140695040,    61917364224;
  9: 5559060566555523, 27177629436493668, 60394732081097040, 80526309441462720, 71578941725744640, 44538008184907776, 19794670304403456, 6284022318858240, 1396449404190720, 206881393213440, 18389457174528, 743008370688;

The row polynomials satisfy P(n,x) = (27+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 729 + 648*x + 144*x^2.

P[0, x_] := 729 + 648*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 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] = 729 + 648*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (27 + 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 (−∞, -1]
  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), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (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
• 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) (above)

Related ?

Same fingerprint ?

• A038328 → basis A013609 → basis A013609
• A013610 → basis A013610
• A038319 → basis A013611 → x ↦ x/(x+1)
• A013610 → D → basis A013610
• A038279 → basis A013611
• basis-map coefficients of A112227 → basis A013610 → basis A013610
• basis-map coefficients of A026729 → basis A013610 → basis A013610
• A038271 → basis A013610 → x ↦ x/(x+1)
• A013610 → basis A013610 → D
• basis-map coefficients of A030528, shifted → basis A013610 → basis A013610
• A038334 → basis A013609 → basis A007318
• A013610 → basis A013611 → basis A013610
• A013622 → basis A013609 → x ↦ x/(x+1)
• basis-map coefficients of A026729, shifted → basis A013610 → basis A013610
• A130123 → basis A013610 → basis A013610
• A038279 → basis A013611 → D
• A136572 → 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.