A034870 → basis A013609 → (θ+1)²

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Submission text ?

Triangle begins:
  0:      1;
  1:      4,       32,        36;
  2:     16,      256,       864,       1024,         400;
  3:     64,     1536,      8640,      20480,       24000,       13824,         3136;
  4:    256,     8192,     64512,     229376,      448000,      516096,       351232,       131072,        20736;
  5:   1024,    40960,    414720,    1966080,     5376000,     9289728,     10536960,      7864320,      3732480,       1024000,        123904;
  6:   4096,   196608,   2433024,   14417920,    50688000,   116785152,    185450496,    207618048,    164229120,      90112000,      32710656,       7077888,       692224;
  7:  16384,   917504,  13418496,   95420416,   410009600,  1180827648,   2410856448,   3598712832,   3985293312,    3280076800,    1984446464,     858783744,    251969536,     44957696,      3686400;
  8:  65536,  4194304,  70778880,  587202560,  2981888000, 10305404928,  25715802112,  47982837760,  68319313920,   74973184000,   63502286848,   41221619712,  20157562880,   7193231360,   1769472000,   268435456,    18939904;
  9: 262144, 18874368, 360972288, 3422552064, 20054016000, 80857792512, 238455619584, 533918121984, 929142669312, 1274544128000, 1387978555392, 1201315774464, 822428565504, 440225759232, 180486144000, 54760833024, 11591221248, 1528823808, 94633984;

The row polynomials satisfy (1+(6*n-4)*x+(4*n^2-4*n+1)*x^2)*P(n,x) = (4+(24*n+16)*x+(16*n^2+64*n+24)*x^2+(32*n^2+56*n+16)*x^3+(16*n^2+16*n+4)*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4 + (24*n + 16)*x + (16*n^2 + 64*n + 24)*x^2 + (32*n^2 + 56*n + 16)*x^3 + (16*n^2 + 16*n + 4)*x^4)*P[n - 1, x])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*x^2)]];
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] = ((4 + (24*n + 16)*x + (16*n^2 + 64*n + 24)*x^2 + (32*n^2 + 56*n + 16)*x^3 + (16*n^2 + 16*n + 4)*x^4)*v[i - 1])/(1 + (6*n - 4)*x + (4*n^2 - 4*n + 1)*x^2)); 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 multiplier sequences
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on multiplier sequences
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

• Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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 ?

• A034870 → (θ+1)²
• A139548 → (θ+1)²
• A139548 → basis A013609 → (θ+1)²

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.