A038271 → basis A013611 → x ↦ x/(x+1)

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Submission text ?

Triangle begins:
  0:          1;
  1:         12,           32;
  2:        144,          768,          1024;
  3:       1728,        13824,         36864,         32768;
  4:      20736,       221184,        884736,       1572864,        1048576;
  5:     248832,      3317760,      17694720,      47185920,       62914560,       33554432;
  6:    2985984,     47775744,     318504960,    1132462080,     2264924160,     2415919104,      1073741824;
  7:   35831808,    668860416,    5350883328,   23781703680,    63417876480,   101468602368,     90194313216,     34359738368;
  8:  429981696,   9172942848,   85614133248,  456608710656,  1522029035520,  3246995275776,   4329327034368,   3298534883328,   1099511627776;
  9: 5159780352, 123834728448, 1320903770112, 8218956791808, 32875827167232, 87668872445952, 155855773237248, 178120883699712, 118747255799808, 35184372088832;

The row polynomials satisfy (1+x)*P(n,x) = (12+44*x+32*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((12 + 44*x + 32*x^2)*P[n - 1, x])/(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] = ((12 + 44*x + 32*x^2)*v[i - 1])/(1 + 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 [-1, 0]
  by Affine, reversal and Möbius substitutions (elementary)
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Affine, reversal and Möbius substitutions (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

• Affine, reversal and Möbius substitutions (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 ?

Candidate equalities ?

• A038208 → basis A013611 → basis A013611 15 rows; no all-index equality certificate was found

Same fingerprint ?

• A013622 → x ↦ x/(x+1) → D
• Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
• A013609 → basis A013610 → basis A013611
• A038329 → basis A013610 → basis A007318
• A027466 → basis A013610 → D
• A013622 → x ↦ x/(x+1)
• A178821 → basis A013611 → basis A013611
• A178820 → basis A013611 → basis A013611
• A038226 → D
• A130123 → basis A013609 → basis A013611
• A120909 → basis A013611
• A038328 → x ↦ x+1 → basis A013610
• A136572 → basis A013609 → basis A013611
• A038327 → basis A013610 → basis A013609
• A028326 → basis A013611 → basis A013611
• basis-map coefficients of A026729, shifted → basis A013609 → basis A013611
• basis-map coefficients of A112227 → basis A013609 → basis A013611
• A038281 → D
• A013612 → basis A013609 → x ↦ x/(x+1)
• A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
• A013609 → D → basis A013611
• basis-map coefficients of A026729 → basis A013609 → basis A013611
• A027466 → basis A013610
• basis-map coefficients of A030528, shifted → basis A013609 → basis A013611
• A120909 → basis A013610 → basis A013611

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.