A038284 → basis A013610 → basis A013610

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Submission text ?

Triangle begins:
  0:              1024,               3456,                2916;
  1:             32768,             165888,              279936,               157464;
  2:           1048576,            7077888,            17915904,             20155392,              8503056;
  3:          33554432,          283115520,           955514880,           1612431360,           1360488960,             459165024;
  4:        1073741824,        10871635968,         45864714240,         103195607040,         130606940160,           88159684608,           24794911296;
  5:       34359738368,       405874409472,       2054739197952,        5778953994240,        9751984865280,         9873884676096,         5554060130304,         1338925209984;
  6:     1099511627776,     14843406974976,      87668872445952,      295882444505088,      624127031377920,       842571492360192,       710919696678912,       342764853755904,        72301961339136;
  7:    35184372088832,    534362651099136,    3606947894919168,    14202357336244224,    35949717007368192,     60665147449933824,     68248290881175552,     49358138940850176,     20822964865671168,      3904305912313344;
  8:  1125899906842624,  18999560927969280,  144277915796766720,   649250621085450240,  1917318240392970240,   3882569436795764736,   5459863270494044160,   5264868153690685440,   3331674378507386880,   1249377891940270080,   210832519264920576;
  9: 36028797018963968, 668784544664518656, 5642869595606876160, 28567027327759810560, 96413717231189360640, 227777406958684864512, 384374374242780708864, 463308397524780318720, 390916460411533393920, 219890508981487534080, 74213046781252042752, 11384956040305711104;

The row polynomials satisfy P(n,x) = (32+54*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 3456*x + 2916*x^2.

P[0, x_] := 1024 + 3456*x + 2916*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 54*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1024 + 3456*x + 2916*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 54*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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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
        • 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)*11^(i-j)*7^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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
            • 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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
            • ↑ Geometric basis map (elementary) (above)
            • 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)*2^j.?
                • Geometric basis map (elementary) polytool ? step ›

                  • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^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
                    • 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) (above)

Related ?

Same fingerprint ?

• A038322 → basis A007318 → basis A013609
• A038226 → basis A013609 → x ↦ x+1
• A038335 → 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.