A038338 → basis A013610 → basis A013609

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Submission text ?

Triangle begins:
  0:                 3600,                  8640,                   5184;
  1:               216000,                777600,                 933120,                  373248;
  2:             12960000,              62208000,              111974400,                89579520,                26873856;
  3:            777600000,            4665600000,            11197440000,             13436928000,              8062156800,              1934917632;
  4:          46656000000,          335923200000,          1007769600000,           1612431360000,           1451188224000,            696570347520,            139314069504;
  5:        2799360000000,        23514624000000,         84652646400000,         169305292800000,         203166351360000,         146279772979200,          58511909191680,          10030613004288;
  6:      167961600000000,      1612431360000000,       6772211712000000,       16253308108800000,       24379962163200000,       23404763676672000,       14042858206003200,        4814694242058240,         722204136308736;
  7:    10077696000000000,    108839116800000000,     522427760640000000,     1462797729792000000,     2633035913625600000,     3159643096350720000,     2527714477080576000,     1299967445355724800,      389990233606717440,       51998697814228992;
  8:   604661760000000000,   7255941120000000000,   39182082048000000000,   125382662553600000000,   263303591362560000000,   379157171562086400000,   379157171562086400000,   259993489071144960000,   116997070082015232000,    31199218688537395200,    3743906242624487424;
  9: 36279705600000000000, 478892113920000000000, 2873352683520000000000, 10344069660672000000000, 24825767185612800000000, 41707288871829504000000, 50048746646195404800000, 42898925696738918400000, 25739355418043351040000, 10295742167217340416000, 2470978120132161699840, 269561249468963094528;

The row polynomials satisfy P(n,x) = (60+72*x)*P(n-1,x) for n >= 1, with P(0,x) = 3600 + 8640*x + 5184*x^2.

P[0, x_] := 3600 + 8640*x + 5184*x^2;
P[n_, x_] := P[n, x] = Expand[(60 + 72*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 3600 + 8640*x + 5184*x^2; for(n = 1, N, my(i = n + 1); v[i] = (60 + 72*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)*12^(i-j)*12^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^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 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Liu–Wang theorem (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• A038329 → x ↦ x+1 → x ↦ x/(x+1)
• A013612 → x ↦ x+1
• A038334 → basis A013610 → reverse
• A178821 → basis A013610 → basis A013609
• A038255 → basis A013610 → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• A028326 → basis A013610 → basis A013609
• A013612 → x ↦ x+1 → D
• A013612 → x ↦ x+1 → reverse
• A038271 → basis A013609 → D
• A156992 → basis A013610 → basis A013609
• A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
• A038259 → D
• A085880 → basis A013610 → basis A013609
• A038291 → basis A013609 → x ↦ x/(x+1)
• A178820 → basis A013610 → basis A013609
• A120910 → basis A013609 → basis A013610
• A196347 → basis A013610 → basis A013609
• A038220 → basis A013610
• A038220 → basis A013610 → D
• A178822 → basis A013610 → basis A013609
• A038334 → basis A013610 → D
• A038314 → D
• A038271 → basis A013609
• A147716 → basis A013610 → x ↦ x/(x+1)
• A038221 → basis A013610 → basis A013609
• A038334 → basis A013610
• A038336 → D

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.