A038329 → basis A013611 → D

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Submission text ?

Triangle begins:
  0:            360,             288;
  1:           8100,           12960,             5184;
  2:         162000,          388800,           311040,            82944;
  3:        3037500,         9720000,         11664000,          6220800,          1244160;
  4:       54675000,       218700000,        349920000,        279936000,        111974400,         17915904;
  5:      956812500,      4592700000,       9185400000,       9797760000,       5878656000,       1881169920,        250822656;
  6:    16402500000,     91854000000,     220449600000,     293932800000,     235146240000,     112870195200,      30098718720,       3439853568;
  7:   276792187500,   1771470000000,    4960116000000,    7936185600000,    7936185600000,    5079158784000,    2031663513600,     464380231680,     46438023168;
  8:  4613203125000,  33215062500000,  106288200000000,  198404640000000,  238085568000000,  190468454400000,  101583175680000,   34828517376000,   6965703475200,    619173642240;
  9: 76117851562500, 608942812500000, 2192194125000000, 4676680800000000, 6547353120000000, 6285458995200000, 4190305996800000, 1915568455680000, 574670536704000, 102163650969600, 8173092077568;

The row polynomials satisfy (n+1)*P(n,x) = (15*n+30+(12*n+24)*x)*P(n-1,x) for n >= 1, with P(0,x) = 360 + 288*x.

P[0, x_] := 360 + 288*x;
P[n_, x_] := P[n, x] = Expand[Together[((15*n + 30 + (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] = 360 + 288*x; for(n = 1, N, my(i = n + 1); v[i] = ((15*n + 30 + (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 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 ?

Same fingerprint ?

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