A038330 → basis A013610 → D

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Submission text ?

Triangle begins:
  0:             384,              288;
  1:            9216,            13824,             5184;
  2:          196608,           442368,           331776,            82944;
  3:         3932160,         11796480,         13271040,          6635520,          1244160;
  4:        75497472,        283115520,        424673280,        318504960,        119439360,         17915904;
  5:      1409286144,       6341787648,      11890851840,      11890851840,       6688604160,       2006581248,        250822656;
  6:     25769803776,     135291469824,     304405807104,     380507258880,     285380444160,     128421199872,      32105299968,       3439853568;
  7:    463856467968,    2783138807808,    7305739370496,   10958609055744,   10273695989760,    6164217593856,    2311581597696,     495338913792,     46438023168;
  8:   8246337208320,   55662776156160,  166988328468480,  292229574819840,  328758271672320,  246568703754240,  123284351877120,   39627113103360,   7430083706880,    619173642240;
  9: 145135534866432, 1088516511498240, 3673743226306560, 7347486452613120, 9643575969054720, 8679218372149248, 5424511482593280, 2324790635397120, 653847366205440, 108974561034240, 8173092077568;

The row polynomials satisfy (n+1)*P(n,x) = (16*n+32+(12*n+24)*x)*P(n-1,x) for n >= 1, with P(0,x) = 384 + 288*x.

P[0, x_] := 384 + 288*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 32 + (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] = 384 + 288*x; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 32 + (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_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^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 ?

• A038284 → reverse
• Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
• A013610 → basis A007318 → D
• A013610 → basis A013611 → x ↦ x+1
• basis-map coefficients of A026729 → basis A013610 → basis A007318
• A038334 → basis A013609 → basis A013610
• A178820 → basis A013611 → basis A013609
• A178822 → basis A013611 → basis A013609
• A027465 → x ↦ x/(x+1)
• A038284 → D
• A130123 → basis A013609 → basis A013609
• A356546 → basis A013611 → basis A013609
• A027465 → D → basis A013610
• A038330 → basis A013610
• basis-map coefficients of A026729, shifted → basis A013610 → x ↦ x+1
• A135065 → x ↦ x+1 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013610 → x ↦ x+1
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013609
• A356546 → x ↦ x+1 → basis A013611
• A038329 → basis A013611 → basis A013610
• A013609 → basis A013609 → D
• A038221 → basis A013611 → basis A013609
• A038335 → reverse
• A120910 → basis A013611
• A178820 → basis A007318 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013609
• A130123 → basis A013610 → x ↦ x+1
• A038208 → basis A013611 → basis A013609
• A038330 → x ↦ x/(x+1)
• A094305 → x ↦ x+1 → basis A013611
• A120909 → basis A013610 → basis A013609
• A120910 → basis A013611 → reverse
• A038329 → basis A013610 → basis A013609
• A038330 → basis A013610 → reverse
• A120910 → basis A013611 → D
• A136572 → basis A013610 → basis A007318
• A013609 → basis A013609
• A085880 → basis A013611 → basis A013609
• A178822 → x ↦ x+1 → basis A013611
• A038335 → D
• A038243 → basis A013609 → x ↦ x/(x+1)
• A085880 → x ↦ x+1 → basis A013611
• A028326 → basis A007318 → basis A013611
• basis-map coefficients of A112227 → basis A013609 → basis A013609
• A028326 → basis A013611 → basis A013609
• A178821 → basis A007318 → basis A013611
• A013610 → basis A007318
• basis-map coefficients of A112227 → basis A013610 → basis A007318
• A038279 → basis A013610 → x ↦ x/(x+1)
• A196347 → x ↦ x+1 → basis A013611
• A178821 → basis A013611 → basis A013609
• A003506 → x ↦ x+1 → basis A013611
• A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
• basis-map coefficients of A026729 → basis A013609 → basis A013609
• A013609 → D → basis A013609
• A120910 → D → basis A013611
• A156992 → basis A013611 → basis A013609
• A085881 → basis A013611 → basis A013609
• A196347 → basis A013611 → 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.