A034870 → basis A013609 → basis A013611

Matches ?

No OEIS matches.

Submission text ?

Triangle begins:
  0:           1;
  1:          16,            64,             64;
  2:         256,          2048,           6144,            8192,             4096;
  3:        4096,         49152,         245760,          655360,           983040,            786432,            262144;
  4:       65536,       1048576,        7340032,        29360128,         73400320,         117440512,         117440512,           67108864,           16777216;
  5:     1048576,      20971520,      188743680,      1006632960,       3523215360,        8455716864,       14092861440,        16106127360,        12079595520,          5368709120,          1073741824;
  6:    16777216,     402653184,     4429185024,     29527900160,     132875550720,      425201762304,      992137445376,      1700807049216,      2126008811520,       1889785610240,       1133871366144,        412316860416,         68719476736;
  7:   268435456,    7516192768,    97710505984,    781684047872,    4299262263296,    17197049053184,    51591147159552,    117922622078976,    206364588638208,     275152784850944,     275152784850944,     200111116255232,     100055558127616,      30786325577728,       4398046511104;
  8:  4294967296,  137438953472,  2061584302080,  19241453486080,  125069447659520,   600333348765696,  2201222278807552,   6289206510878720,  14150714649477120,   25156826043514880,   35219556460920832,   38421334321004544,   32017778600837120,   19703248369745920,    8444249301319680,    2251799813685248,    281474976710656;
  9: 68719476736, 2473901162496, 42056319762432, 448600744132608, 3364505580994560, 18841231253569536, 81645335432134656, 279926864338747392, 769798876931555328, 1710664170959011840, 3079195507726221312, 4478829829419958272, 5225301467656617984, 4823355200913801216, 3445253714938429440, 1837468647967162368, 689050742987685888, 162129586585337856, 18014398509481984;

The row polynomials satisfy P(n,x) = (16+64*x+64*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[(16 + 64*x + 64*x^2)*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] = (16 + 64*x + 64*x^2)*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_A013609 of Even-numbered rows of Pascal's triangle.?
    • Geometric basis map (elementary) polytool ? step ›

      • from Even-numbered rows of Pascal's triangle.?
        • 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 Even-numbered rows of Pascal's triangle.?
            • ↑ Parent-row provers of polytool (above)
            • ↑ Iterated real-rootedness preservers (above)
            • Geometric basis map (elementary) polytool ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                • 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 basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                    • ↑ Parent-row provers of polytool (above)
                    • ↑ Iterated real-rootedness preservers (above)
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…?
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
                • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
            • 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
        • ↑ Geometric basis map (elementary) (above)
        • ↑ Affine, reversal and Möbius substitutions (elementary) (above)
        • ↑ Row-hypergeometric rows (multiplier sequences) (above)
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Even-numbered rows of Pascal's triangle.?
        • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A139548 → basis A013611
• basis-map coefficients of A297191, shifted → θ+1 → basis A013609
• A139548 → x ↦ x+1
• A139548 → basis A013609 → basis A007318
• A034870 → basis A013611
• basis-map coefficients of A297191, shifted → (θ+1)² → basis A013609
• A139548 → x ↦ x/(x+1)
• basis-map coefficients of A297191, shifted → basis A013609
• A034870 → x ↦ x/(x+1)
• A034870 → basis A013611 → basis A013610
• A034870 → x ↦ x+1
• A034870 → basis A013609 → x ↦ x+1
• A139548 → x ↦ x+1 → reverse
• basis-map coefficients of A297191, shifted → basis A013609 → basis A013610

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.