A038321 → basis A007318 → x ↦ x+1

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:              625,              350,                49;
  1:            15625,            13125,              3675,              343;
  2:           390625,           437500,            183750,            34300,             2401;
  3:          9765625,         13671875,           7656250,          2143750,           300125,            16807;
  4:        244140625,        410156250,         287109375,        107187500,         22509375,          2521050,          117649;
  5:       6103515625,      11962890625,       10048828125,       4689453125,       1313046875,        220591875,        20588575,          823543;
  6:     152587890625,     341796875000,      334960937500,     187578125000,      65652343750,      14706125000,      2058857500,       164708600,        5764801;
  7:    3814697265625,    9613037109375,    10766601562500,    7034179687500,    2954355468750,     827219531250,    154414312500,     18529717500,     1297080225,      40353607;
  8:   95367431640625,  267028808593750,   336456298828125,  251220703125000,  123098144531250,   41360976562500,   9650894531250,   1544143125000,   162135028125,   10088401750,   282475249;
  9: 2384185791015625, 7343292236328125, 10280609130859375, 8635711669921875, 4835998535156250, 1895711425781250, 530799199218750, 106159839843750, 14862377578125, 1387155240625, 77680693475, 1977326743;

The row polynomials satisfy P(n,x) = (25+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 625 + 350*x + 49*x^2.

P[0, x_] := 625 + 350*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[(25 + 7*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 625 + 350*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (25 + 7*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 (−∞, -1]
  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), Geometric basis map (elementary)
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Affine, reversal and Möbius substitutions (elementary)
• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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
        • 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
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
        • ↑ Liu–Wang theorem (above)
        • ↑ Parent-row provers of polytool (above)
        • ↑ Iterated real-rootedness preservers (above)
• Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

  • from basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.?
    • ↑ Geometric basis map (elementary) (above)
    • ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

• A038277 → x ↦ x/(x+1) → x ↦ x/(x+1)

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.