A038331 → D

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Submission text ?

Triangle begins:
  0:           120,             50;
  1:          2160,           1800,            375;
  2:         34560,          43200,          18000,           2500;
  3:        518400,         864000,         540000,         150000,          15625;
  4:       7464960,       15552000,       12960000,        5400000,        1125000,          93750;
  5:     104509440,      261273600,      272160000,      151200000,       47250000,        7875000,        546875;
  6:    1433272320,     4180377600,     5225472000,     3628800000,     1512000000,      378000000,      52500000,      3125000;
  7:   19349176320,    64497254400,    94058496000,    78382080000,    40824000000,    13608000000,    2835000000,    337500000,     17578125;
  8:  257989017600,   967458816000,  1612431360000,  1567641600000,   979776000000,   408240000000,  113400000000,  20250000000,   2109375000,    97656250;
  9: 3405455032320, 14189395968000, 26605117440000, 29561241600000, 21555072000000, 10777536000000, 3742200000000, 891000000000, 139218750000, 12890625000, 537109375;

The row polynomials satisfy (n+1)*P(n,x) = (12*n+24+(5*n+10)*x)*P(n-1,x) for n >= 1, with P(0,x) = 120 + 50*x.

P[0, x_] := 120 + 50*x;
P[n_, x_] := P[n, x] = Expand[Together[((12*n + 24 + (5*n + 10)*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] = 120 + 50*x; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 24 + (5*n + 10)*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)*12^(i-j)*5^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

Related ?

Same fingerprint ?

• A013620 → basis A013611
• A013611 → D → basis A013610
• A178820 → basis A013610 → basis A013611
• basis-map coefficients of A112227 → basis A013611 → basis A013610
• A130123 → basis A013611 → basis A013610
• A136572 → basis A013611 → basis A013610
• A178821 → basis A013610 → basis A013611
• A013620 → basis A013611 → D
• A178822 → basis A013610 → basis A013611
• A085880 → basis A013610 → basis A013611
• A038331 → reverse
• A028326 → basis A013610 → basis A013611
• basis-map coefficients of A026729 → basis A013611 → basis A013610
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
• A038271 → basis A007318
• A038271 → basis A007318 → 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.