A013612 → basis A013611 → D+1

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Submission text ?

Triangle begins:
  0:         276,         1040,           400;
  1:        2376,        16560,         31200,           8000;
  2:       18576,       190080,        662400,         832000,          160000;
  3:      137376,      1857600,       9504000,       22080000,        20800000,          3200000;
  4:      979776,     16485120,     111456000,      380160000,       662400000,        499200000,         64000000;
  5:     6811776,    137168640,    1153958400,     5201280000,     13305600000,      18547200000,      11648000000,       1280000000;
  6:    46469376,   1089884160,   10973491200,    61544448000,    208051200000,     425779200000,     494592000000,     266240000000,       25600000000;
  7:   312408576,   8364487680,   98089574400,   658409472000,   2769500160000,    7489843200000,   12773376000000,   12718080000000,     5990400000000,     512000000000;
  8:  2076005376,  62481715200,  836448768000,  6539304960000,  32920473600000,  110780006400000,  249661440000000,  364953600000000,   317952000000000,  133120000000000,   10240000000000;
  9: 13665355776, 456721182720, 6872988672000, 61339576320000, 359661772800000, 1448500838400000, 4061933568000000, 7846502400000000, 10036224000000000, 7772160000000000, 2928640000000000, 204800000000000;

The row polynomials satisfy (10*n+13+10*x)*P(n,x) = (60*n+138+(200*n+520)*x+200*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 276 + 1040*x + 400*x^2.

P[0, x_] := 276 + 1040*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((60*n + 138 + (200*n + 520)*x + 200*x^2)*P[n - 1, x])/(10*n + 13 + 10*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 276 + 1040*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((60*n + 138 + (200*n + 520)*x + 200*x^2)*v[i - 1])/(10*n + 13 + 10*x)); 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 of coefficients in expansion of (1+5x)^n.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle of coefficients in expansion of (1+5x)^n.?
        • 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

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.