A038271 → basis A013611 → basis A013609

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Submission text ?

Triangle begins:
  0:              1;
  1:             32,              40;
  2:           1024,            2560,             1600;
  3:          32768,          122880,           153600,            64000;
  4:        1048576,         5242880,          9830400,          8192000,           2560000;
  5:       33554432,       209715200,        524288000,        655360000,         409600000,         102400000;
  6:     1073741824,      8053063680,      25165824000,      41943040000,       39321600000,       19660800000,        4096000000;
  7:    34359738368,    300647710720,    1127428915200,    2348810240000,     2936012800000,     2202009600000,      917504000000,     163840000000;
  8:  1099511627776,  10995116277760,   48103633715200,  120259084288000,   187904819200000,   187904819200000,   117440512000000,   41943040000000,    6553600000000;
  9: 35184372088832, 395824185999360, 1979120929996800, 5772436045824000, 10823317585920000, 13529146982400000, 11274289152000000, 6039797760000000, 1887436800000000, 262144000000000;

The row polynomials satisfy P(n,x) = (32+40*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(32 + 40*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (32 + 40*x)*v[i - 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 Geometric basis map (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^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 ?

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

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.