A038314 → D

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Submission text ?

Triangle begins:
  0:           240,            288;
  1:          3600,           8640,           5184;
  2:         48000,         172800,         207360,           82944;
  3:        600000,        2880000,        5184000,         4147200,         1244160;
  4:       7200000,       43200000,      103680000,       124416000,        74649600,        17915904;
  5:      84000000,      604800000,     1814400000,      2903040000,      2612736000,      1254113280,       250822656;
  6:     960000000,     8064000000,    29030400000,     58060800000,     69672960000,     50164531200,     20065812480,      3439853568;
  7:   10800000000,   103680000000,   435456000000,   1045094400000,   1567641600000,   1504935936000,    902961561600,    309586821120,     46438023168;
  8:  120000000000,  1296000000000,  6220800000000,  17418240000000,  31352832000000,  37623398400000,  30098718720000,  15479341056000,   4643802316800,   619173642240;
  9: 1320000000000, 15840000000000, 85536000000000, 273715200000000, 574801920000000, 827714764800000, 827714764800000, 567575838720000, 255409127424000, 68109100646400, 8173092077568;

The row polynomials satisfy (n+1)*P(n,x) = (10*n+20+(12*n+24)*x)*P(n-1,x) for n >= 1, with P(0,x) = 240 + 288*x.

P[0, x_] := 240 + 288*x;
P[n_, x_] := P[n, x] = Expand[Together[((10*n + 20 + (12*n + 24)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 240 + 288*x; for(n = 1, N, my(i = n + 1); v[i] = ((10*n + 20 + (12*n + 24)*x)*v[i - 1])/(n + 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 Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.?
    • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    • 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
    • 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)*10^j.?
        • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
        • 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
        • Parent-row provers of polytool polytool ? step › direct check
        • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.?
            • ↑ Liu–Wang theorem (above)
            • ↑ Row-hypergeometric rows (multiplier sequences) (above)
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
        • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

• A156992 → basis A013610 → basis A013609
• A178821 → basis A013610 → basis A013609
• A038221 → basis A013610 → basis A013609
• A038259 → D
• A178822 → basis A013610 → basis A013609
• A038220 → basis A013610
• A038271 → basis A013609
• A013612 → x ↦ x+1
• A038255 → basis A013610 → basis A013611
• A147716 → basis A013610 → x ↦ x/(x+1)
• A038334 → basis A013610 → D
• A013612 → x ↦ x+1 → reverse
• A178820 → basis A013610 → basis A013609
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
• A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
• A038220 → basis A013610 → D
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
• A038334 → basis A013610
• A038329 → x ↦ x+1 → x ↦ x/(x+1)
• A028326 → basis A013610 → basis A013609
• A120910 → basis A013609 → basis A013610
• A013612 → x ↦ x+1 → D
• A038336 → D
• A085880 → basis A013610 → basis A013609
• A038334 → basis A013610 → reverse
• A038338 → basis A013610 → basis A013609
• A038271 → basis A013609 → D
• A038291 → basis A013609 → x ↦ x/(x+1)
• A196347 → basis A013610 → 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.