basis-map coefficients of A026729 → basis A013611

OEIS matches ?

• A074701: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A038231: reversed 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 = 0 ?
• A000351: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
• A384565: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A013611: 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 = 0 ?

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:
   1: 1;
   2: 1,  4;
   3: 1,  8,  16;
   4: 1, 12,  48,   64;
   5: 1, 16,  96,  256,   256;
   6: 1, 20, 160,  640,  1280,   1024;
   7: 1, 24, 240, 1280,  3840,   6144,   4096;
   8: 1, 28, 336, 2240,  8960,  21504,  28672,  16384;
   9: 1, 32, 448, 3584, 17920,  57344, 114688, 131072,  65536;
  10: 1, 36, 576, 5376, 32256, 129024, 344064, 589824, 589824, 262144;

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

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

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

Triangle begins:
  0: 1;
  1: 1,  4;
  2: 1,  8,  16;
  3: 1, 12,  48,   64;
  4: 1, 16,  96,  256,   256;
  5: 1, 20, 160,  640,  1280,   1024;
  6: 1, 24, 240, 1280,  3840,   6144,   4096;
  7: 1, 28, 336, 2240,  8960,  21504,  28672,  16384;
  8: 1, 32, 448, 3584, 17920,  57344, 114688, 131072,  65536;
  9: 1, 36, 576, 5376, 32256, 129024, 344064, 589824, 589824, 262144;

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

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

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

Triangle begins:
  0:      1;
  1:      4,      1;
  2:     16,      8,      1;
  3:     64,     48,     12,      1;
  4:    256,    256,     96,     16,      1;
  5:   1024,   1280,    640,    160,     20,     1;
  6:   4096,   6144,   3840,   1280,    240,    24,    1;
  7:  16384,  28672,  21504,   8960,   2240,   336,   28,   1;
  8:  65536, 131072, 114688,  57344,  17920,  3584,  448,  32,  1;
  9: 262144, 589824, 589824, 344064, 129024, 32256, 5376, 576, 36, 1;

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

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

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + 4*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 >= 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-f93ad378f9476e41
• 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-f93ad378f9476e41
• 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-f93ad378f9476e41

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

• basis_map A013611 e-f93ad378f9476e41

  Geometric basis map (elementary) polytool ?

  • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-6e85c0971e84e272 root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image mobius, shift 0 e-ad67990681a86223

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
        uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-6e85c0971e84e272 (expanded above)

Related sequences ?

Same fingerprint (14)

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

• A038231 → D
• A027467 → basis A013611
• basis-map coefficients of A030528, shifted → basis A013611
• basis-map coefficients of A112227 → basis A013611
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
• A136572 → basis A013611
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
• basis-map coefficients of A026729, shifted → basis A013611
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
• A013611 → D
• A038330 → basis A007318
• A027466 → basis A013609
• A130123 → basis A013611
• basis-map coefficients of A021012 → basis A013611