Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).

OEIS matches ?

• A129683: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
   1:    4,     16,        8;
   2:    8,     72,      144,        48;
   3:   16,    256,     1152,      1536,        384;
   4:   32,    800,     6400,     19200,      19200,        3840;
   5:   64,   2304,    28800,    153600,     345600,      276480,        46080;
   6:  128,   6272,   112896,    940800,    3763200,     6773760,      4515840,        645120;
   7:  256,  16384,   401408,   4816896,   30105600,    96337920,    144506880,      82575360,      10321920;
   8:  512,  41472,  1327104,  21676032,  195084288,   975421440,   2601123840,    3344302080,    1672151040,     185794560;
   9: 1024, 102400,  4147200,  88473600, 1083801600,  7803371520,  32514048000,   74317824000,   83607552000,   37158912000,   3715891200;
  10: 2048, 247808, 12390400, 334540800, 5352652800, 52455997440, 314735984640, 1124057088000, 2248114176000, 2248114176000, 899245670400, 81749606400;

The row polynomials satisfy P(n,x) = (2+(4*n+2)*x)*P(n-1,x) - 2*x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 4 + 16*x + 8*x^2.

P[1, x_] := 4 + 16*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + (4*n + 2)*x)*P[n - 1, x] - 2*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = 4 + 16*x + 8*x^2; for(n = 2, N, my(i = n); v[i] = (2 + (4*n + 2)*x)*v[i - 1] - 2*x^2*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-b9d2d25190d8d499
• Coefficients of one sign. Every nonzero row, n >= 1: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-b9d2d25190d8d499
• Hadamard kernel (Malo). Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-b9d2d25190d8d499
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-b9d2d25190d8d499

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• theorem_check liu_wang e-b9d2d25190d8d499 root: theorem check

  Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

Used in 2 proof steps

• e-f9379b86f8cefc86 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n…
• e-f9379b86f8cefc86 obreschkoff_combination → obreschkoff_combination of Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n…

Related sequences ?

Same fingerprint (2)

Equal up to content, sign, reversal and a small index shift.

• A144084: T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.
• A132393 → basis A008277

References

1. [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