A227450: Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.

OEIS matches ?

• A001861: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A227450: triangle, 55 terms agree, 10 full rows row-aligned (tabl) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 990 terms from n = 1 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:
  0: 2;
  1: 2,    4;
  2: 2,   12,     8;
  3: 2,   28,    48,     16;
  4: 2,   60,   200,    160,      32;
  5: 2,  124,   720,   1040,     480,      64;
  6: 2,  252,  2408,   5600,    4480,    1344,    128;
  7: 2,  508,  7728,  27216,   33600,   17024,   3584,    256;
  8: 2, 1020, 24200, 124320,  222432,  169344,  59136,   9216,   512;
  9: 2, 2044, 74640, 545680, 1360800, 1460928, 752640, 192000, 23040, 1024;

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.

P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] + x*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 - 1, v[i])))};
T(9)

Triangle begins:
   1: 2;
   2: 2,    4;
   3: 2,   12,     8;
   4: 2,   28,    48,     16;
   5: 2,   60,   200,    160,      32;
   6: 2,  124,   720,   1040,     480,      64;
   7: 2,  252,  2408,   5600,    4480,    1344,    128;
   8: 2,  508,  7728,  27216,   33600,   17024,   3584,    256;
   9: 2, 1020, 24200, 124320,  222432,  169344,  59136,   9216,   512;
  10: 2, 2044, 74640, 545680, 1360800, 1460928, 752640, 192000, 23040, 1024;

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

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

P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = (1 + 2*x)*v[i - 1] + x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); 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 >= 0, 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-74361f30d2bc482d
• Coefficients of one sign. Every nonzero row, n >= 0: 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-74361f30d2bc482d
• Hadamard kernel (Malo). Rows P_n, n >= 0, 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-74361f30d2bc482d
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-74361f30d2bc482d
• Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 1] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 1] with m = 0, target (-inf, 0]
  The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
  by the cited steps: e-982f957c1008df8b

Proof chains

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

• basis_map_theorem e-982f957c1008df8b root: theorem check

  Ray basis map theorem [MW08, Thm 2] polytool RealRooted rayBasisMap_maps_realRootedIn_iff ?

• theorem_check liu_wang e-74361f30d2bc482d root: theorem check

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

Used in 9 proof steps

• e-9dc64aa496a51831 obreschkoff_combination → obreschkoff_combination of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >=…
• e-9dc64aa496a51831 obreschkoff_combination → obreschkoff_combination of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >=…
• e-12a33f06274c1738 preserver_image (mobius, shift 0) → mobius_x_over_x_plus_one of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >…
• e-36c887a5cdb807b7 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= …
• e-6dbcd50d3871cc63 preserver_image (derivative_polynomial, shift 0) → derivative of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <=…
• e-88af8bd6d056b310 preserver_image (reversal, shift 0) → reversal of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.
• e-b0ca16d4a4893249 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, …
• e-c012cb3d271aa10d preserver_image (affine, shift 0) → shift_x_plus_one of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <…
• e-ca8ea9a3beb129bc preserver_image (euler_polynomial, shift 0) → theta_plus_one_squared of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= …

Related sequences ?

Same fingerprint (1)

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

• A227450 → reverse

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
2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741