A038330 → basis A013611 → D+1

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Triangle begins:
  0:             768,             1024,              256;
  1:           16384,            36864,            24576,              4096;
  2:          327680,          1048576,          1179648,            524288,             65536;
  3:         6291456,         26214400,         41943040,          31457280,          10485760,           1048576;
  4:       117440512,        603979776,       1258291200,        1342177280,         754974720,         201326592,          16777216;
  5:      2147483648,      13153337344,      33822867456,       46976204800,       37580963840,       16911433728,        3758096384,         268435456;
  6:     38654705664,     274877906944,     841813590016,     1443109011456,     1503238553600,      962072674304,      360777252864,       68719476736,        4294967296;
  7:    687194767360,    5566277615616,   19791209299968,    40407052320768,    51951924412416,    43293270343680,    23089744183296,     7421703487488,     1236950581248,      68719476736;
  8:  12094627905536,  109951162777600,  445302209249280,  1055531162664960,  1616282092830720,  1662461581197312,  1154487209164800,   527765581332480,   148434069749760,   21990232555520,   1099511627776;
  9: 211106232532992, 2128654511374336, 9675702324428800, 26124396275957760, 46443371157258240, 56893129667641344, 48765539715121152, 29027106973286400, 11610842789314560, 2902710697328640, 387028092977152, 17592186044416;

The row polynomials satisfy (n+2+x)*P(n,x) = (16*n+48+(16*n+64)*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 768 + 1024*x + 256*x^2.

P[0, x_] := 768 + 1024*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 48 + (16*n + 64)*x + 16*x^2)*P[n - 1, x])/(n + 2 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 768 + 1024*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 48 + (16*n + 64)*x + 16*x^2)*v[i - 1])/(n + 2 + 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 (−∞, 0]
  by Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • 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
            • 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
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• basis-map coefficients of A113214, shifted → basis A013609 → D+1
• basis-map coefficients of A112227 → x ↦ x+1 → θ+1
• A134346 → θ+1
• A131431 → x ↦ x+1 → θ+1
• A027465 → basis A013611 → θ+1
• A124860 → D+1
• basis-map coefficients of A113214, shifted → D → θ+1
• A038208 → D → θ+1
• A178820 → D+1
• A216973 → θ+1
• basis-map coefficients of A127647 → θ+1
• A038338 → basis A013609 → D+1
• A085880 → basis A013609 → D+1
• A135838 → D+1
• A085880 → basis A013609 → θ+1
• A085881 → θ+1
• A120910 → basis A013610 → D+1
• A038208 → D+1
• A178821 → θ+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A178820 → θ+1
• A356546 → basis A013609 → D+1
• A356546 → θ+1
• A038338 → θ+1
• A131431 → D+1 → basis A007318
• A028326 → θ+1
• A027465 → basis A013611 → D+1
• A178821 → basis A013609 → D+1
• A156992 → θ+1
• basis-map coefficients of A113214, shifted → θ+1
• A085880 → D → θ+1
• A038330 → basis A013611 → θ+1
• basis-map coefficients of A127647 → D+1
• A094305 → θ+1
• A124860 → θ+1
• A038325 → basis A013609 → θ+1
• A085880 → D+1
• A135838 → θ+1
• A178822 → θ+1
• A085881 → D+1
• A093375 → D
• A038325 → basis A013609 → D+1
• A028326 → basis A013609 → D+1
• A196347 → D+1
• A356546 → D+1
• basis-map coefficients of A026729, shifted → basis A007318 → D+1
• A178820 → basis A013609 → D+1
• A028326 → D+1
• A038338 → D+1
• A038325 → D+1
• A178821 → D+1
• A135065 → D+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• A344565 → θ+1
• A178822 → D+1
• A178822 → basis A013609 → D+1
• A093375 → D → D
• A085880 → θ+1
• basis-map coefficients of A026729, shifted → basis A007318 → θ+1
• basis-map coefficients of A026729 → basis A007318 → θ+1
• A038221 → θ+1
• A028326 → D → θ+1
• basis-map coefficients of A113214, shifted → D+1
• A156992 → D+1
• A178821 → basis A013609 → θ+1
• A134346 → D+1
• A038208 → D → D+1
• A196347 → basis A013609 → D+1
• A094305 → D+1
• A376827 → D+1
• A038221 → basis A013609 → θ+1
• basis-map coefficients of A112227 → x ↦ x+1 → D+1
• A196347 → basis A013609 → θ+1
• A038221 → D+1
• A038221 → basis A013609 → D+1
• A028326 → basis A013609 → θ+1
• A178820 → basis A013609 → θ+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A028326 → D → D+1
• A038338 → basis A013609 → θ+1
• A003506 → D → θ+1
• A135065 → θ+1
• A344565 → D+1
• A156992 → basis A013609 → θ+1
• basis-map coefficients of A026729 → basis A007318 → D+1
• basis-map coefficients of A113214, shifted → D → D+1
• A178822 → basis A013609 → θ+1
• A038325 → θ+1
• A085880 → D → D+1
• A156992 → basis A013609 → D+1
• basis-map coefficients of A113214, shifted → basis A013609 → θ+1
• A376827 → θ+1
• A085881 → basis A013609 → D+1
• A038208 → θ+1
• A356546 → basis A013609 → θ+1
• A085881 → basis A013609 → θ+1
• A196347 → θ+1
• A131431 → x ↦ x+1 → D+1
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A120910 → basis A013610 → θ+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.