A297191: Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.

OEIS matches ?

• A297191: triangle, 58 terms agree, 7 full rows row lengths from the atlas (tabf) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 1024 terms from n = 0 ?
• A297191: reversed triangle, 58 terms agree, 7 full rows row lengths from the atlas (tabf) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 1024 terms from n = 0 ?
• A001653: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 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: 1;
  1: 1,  3,   1;
  2: 1,  7,  13,    7,     1;
  3: 1, 11,  41,   63,    41,     11,      1;
  4: 1, 15,  85,  231,   321,    231,     85,     15,       1;
  5: 1, 19, 145,  575,  1289,   1683,   1289,    575,     145,      19,       1;
  6: 1, 23, 221, 1159,  3649,   7183,   8989,   7183,    3649,    1159,     221,     23,      1;
  7: 1, 27, 313, 2047,  8361,  22363,  40081,  48639,   40081,   22363,    8361,   2047,    313,     27,     1;
  8: 1, 31, 421, 3303, 16641,  56695, 134245, 224143,  265729,  224143,  134245,  56695,  16641,   3303,   421,   31,   1;
  9: 1, 35, 545, 4991, 29961, 124515, 369305, 795455, 1256465, 1462563, 1256465, 795455, 369305, 124515, 29961, 4991, 545, 35, 1;

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

P[0, x_] := 1;
P[1, x_] := 1 + 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x + x^2)*P[n - 1, x] - x^2*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x + x^2)*v[i - 1] - x^2*v[i - 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)

Triangle begins:
  0: 1;
  1: 1,  3,   1;
  2: 1,  7,  13,    7,     1;
  3: 1, 11,  41,   63,    41,     11,      1;
  4: 1, 15,  85,  231,   321,    231,     85,     15,       1;
  5: 1, 19, 145,  575,  1289,   1683,   1289,    575,     145,      19,       1;
  6: 1, 23, 221, 1159,  3649,   7183,   8989,   7183,    3649,    1159,     221,     23,      1;
  7: 1, 27, 313, 2047,  8361,  22363,  40081,  48639,   40081,   22363,    8361,   2047,    313,     27,     1;
  8: 1, 31, 421, 3303, 16641,  56695, 134245, 224143,  265729,  224143,  134245,  56695,  16641,   3303,   421,   31,   1;
  9: 1, 35, 545, 4991, 29961, 124515, 369305, 795455, 1256465, 1462563, 1256465, 795455, 369305, 124515, 29961, 4991, 545, 35, 1;

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

P[0, x_] := 1;
P[1, x_] := 1 + 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x + x^2)*P[n - 1, x] - x^2*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x + x^2)*v[i - 1] - x^2*v[i - 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)

Triangle begins:
  0: 1;
  1: 1,  3,   1;
  2: 1,  7,  13,    7,     1;
  3: 1, 11,  41,   63,    41,     11,      1;
  4: 1, 15,  85,  231,   321,    231,     85,     15,       1;
  5: 1, 19, 145,  575,  1289,   1683,   1289,    575,     145,      19,       1;
  6: 1, 23, 221, 1159,  3649,   7183,   8989,   7183,    3649,    1159,     221,     23,      1;
  7: 1, 27, 313, 2047,  8361,  22363,  40081,  48639,   40081,   22363,    8361,   2047,    313,     27,     1;
  8: 1, 31, 421, 3303, 16641,  56695, 134245, 224143,  265729,  224143,  134245,  56695,  16641,   3303,   421,   31,   1;
  9: 1, 35, 545, 4991, 29961, 124515, 369305, 795455, 1256465, 1462563, 1256465, 795455, 369305, 124515, 29961, 4991, 545, 35, 1;

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+4*x+x^2)*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x + x^2.

P[0, x_] := 1;
P[1, x_] := 1 + 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x + x^2)*P[n - 1, x] - x^2*P[n - 2, x]];
Table[Reverse[CoefficientList[P[n, x], x]], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x + x^2)*v[i - 1] - x^2*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1564556493/268435456, -46056243/268435456]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-80180a24956ab79f, e-a8f2d1946d8f3a6f
• 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-054f97e0823e3ab6, e-80180a24956ab79f, e-912230b718f958a6, e-a8f2d1946d8f3a6f
• 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-054f97e0823e3ab6, e-80180a24956ab79f, e-912230b718f958a6, e-a8f2d1946d8f3a6f
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-054f97e0823e3ab6, e-912230b718f958a6

Proof chains

5 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 A008288 e-054f97e0823e3ab6

  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

  • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    no recorded property covers this requirement
• basis_map A008288 e-80180a24956ab79f

  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

  • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    no recorded property covers this requirement
• basis_map A008288 e-912230b718f958a6

  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

  • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    no recorded property covers this requirement
• basis_map A008288 e-a8f2d1946d8f3a6f

  Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

  • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
    no recorded property covers this requirement
• definition_equality e-7ee38b0846ecb43c root: theorem check

  Ore-algebra identity of two definitions polytool ?

Used in 1 proof step

• e-313e6d9276f9a630 gamma_polynomial (scale 1, shift 0) → Triangular array binomial(2*n-k, k), k=0..n, n >= 0.

Related sequences ?

No candidate equalities or related sequences.