A165193 Eigensequence of triangle A038208.
1, 4, 28, 352, 7888, 319168, 23833792, 3359617024, 911281182976
Offset: 1
Crossrefs
Cf. A038208.
This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The triangle starts: [ 0] 1 [ 1] 1 1 [ 2] 0 1 1 [ 3] 1 1 1 1 [ 4] 1 3 2 1 1 [ 5] 3 5 5 3 1 1 [ 6] 6 13 10 7 4 1 1 [ 7] 15 29 26 16 9 5 1 1 [ 8] 36 73 61 42 23 11 6 1 1 [ 9] 91 181 157 103 61 31 13 7 1 1 [10] 232 465 398 271 156 83 40 15 8 1 1
RiordanSquare := proc(d, n, exp:=false) local td, M, k, m, u, j;
series(d, x, n+1); td := [seq(coeff(%, x, j), j = 0..n)];
M := Matrix(n); for k from 1 to n do M[k, 1] := td[k] od;
for k from 1 to n-1 do for m from k to n-1 do
M[m+1, k+1] := add(M[j, k]*td[m-j+2], j = k..m) od od;
if exp then u := 1;
for k from 1 to n-1 do u := u * k;
for m from 1 to k do j := `if`(m = 1, u, j/(m-1));
M[k+1, m] := M[k+1, m] * j od od fi;
M end:
RiordanSquare(1 + 2*x/(1 + x + sqrt(1 - 2*x - 3*x^2)), 8);
RiordanSquare[gf_, len_] := Module[{T}, T[n_, k_] := T[n, k] = If[k == 0, SeriesCoefficient[gf, {x, 0, n}], Sum[T[j, k-1] T[n-j, 0], {j, k-1, n-1}]]; Table[T[n, k], {n, 0, len-1}, {k, 0, n}]];
M = RiordanSquare[1 + 2x/(1 + x + Sqrt[1 - 2x - 3x^2]), 12];
M // Flatten (* Jean-François Alcover, Nov 24 2018 *)
# uses[riordan_array from A256893] def riordan_square(gf, len, exp=false): return riordan_array(gf, gf, len, exp) riordan_square(1 + 2*x/(1 + x + sqrt(1 - 2*x - 3*x^2)), 10) # Alternatively, given a list S: def riordan_square_array(S): N = len(S) M = matrix(ZZ, N, N) for n in (0..N-1): M[n, 0] = S[n] for k in (1..N-1): for m in (k..N-1): M[m, k] = sum(M[j, k-1]*S[m-j] for j in (k-1..m-1)) return M riordan_square_array([1, 1, 0, 1, 1, 3, 6, 15, 36]) # Peter Luschny, Apr 03 2020
Array, t(n, k), begins as:
1, 2, 4, 8, 16, 32, 64, ...;
2, 12, 52, 196, 684, 2276, 7340, ...;
4, 52, 416, 2644, 14680, 74652, 357328, ...;
8, 196, 2644, 26440, 220280, 1623964, 10978444, ...;
16, 684, 14680, 220280, 2643360, 27227908, 251195000, ...;
32, 2276, 74652, 1623964, 27227908, 381190712, 4677894984, ...;
64, 7340, 357328, 10978444, 251195000, 4677894984, 74846319744, ...;
Triangle, T(n, k), begins as:
1;
2, 2;
4, 12, 4;
8, 52, 52, 8;
16, 196, 416, 196, 16;
32, 684, 2644, 2644, 684, 32;
64, 2276, 14680, 26440, 14680, 2276, 64;
128, 7340, 74652, 220280, 220280, 74652, 7340, 128;
256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172, 256;
A256890:= func< n,k | (&+[(-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n: j in [0..k]]) >; [A256890(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 18 2022
Table[Sum[(-1)^(k-j)*Binomial[j+3, j] Binomial[n+4, k-j] (j+2)^n, {j,0,k}], {n,0, 9}, {k,0,n}]//Flatten (* Michael De Vlieger, Dec 27 2019 *)
t(n,m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, (m+2)*t(n-1, m) + (n+2)*t(n, m-1)));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", ");); print(););} \\ Michel Marcus, Apr 14 2015
def A256890(n,k): return sum((-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n for j in range(k+1)) flatten([[A256890(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Oct 18 2022
Triangle begins as:
1;
2, 2;
4, 16, 4;
8, 88, 88, 8;
16, 416, 1056, 416, 16;
32, 1824, 9664, 9664, 1824, 32;
64, 7680, 76224, 154624, 76224, 7680, 64;
128, 31616, 549504, 1999232, 1999232, 549504, 31616, 128;
256, 128512, 3739648, 22587904, 39984640, 22587904, 3739648, 128512, 256;
function T(n,k,a,b) if k lt 0 or k gt n then return 0; elif k eq 0 or k eq n then return 1; else return (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b); end if; return T; end function; [T(n,k,2,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 21 2022
A257609:= func< n,k | 2^n*EulerianNumber(n+1, k) >; [A257609(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 17 2025
T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,2,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)
def T(n,k,a,b): # A257609 if (k<0 or k>n): return 0 elif (n==0): return 1 else: return (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b) flatten([[T(n,k,2,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022
Triangle begins as:
1;
2, 2;
4, 20, 4;
8, 132, 132, 8;
16, 748, 2112, 748, 16;
32, 3964, 25124, 25124, 3964, 32;
64, 20364, 256488, 552728, 256488, 20364, 64;
128, 103100, 2398092, 9670840, 9670840, 2398092, 103100, 128;
256, 518444, 21246736, 147146804, 270783520, 147146804, 21246736, 518444, 256;
T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,3,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
def T(n,k,a,b): # A257610 if (k<0 or k>n): return 0 elif (n==0): return 1 else: return (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b) flatten([[T(n,k,3,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022
Triangle begins as:
1;
2, 2;
4, 24, 4;
8, 184, 184, 8;
16, 1216, 3680, 1216, 16;
32, 7584, 53824, 53824, 7584, 32;
64, 46208, 674752, 1507072, 674752, 46208, 64;
128, 278912, 7764096, 33244544, 33244544, 7764096, 278912, 128;
T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,4,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
f(x) = 4*x + 2; T(n, k) = t(n-k, k); t(n, m) = if (!n && !m, 1, if (n < 0 || m < 0, 0, f(m)*t(n-1,m) + f(n)*t(n,m-1))); tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ");); print();); \\ Michel Marcus, May 06 2015
def T(n,k,a,b): # A257612 if (k<0 or k>n): return 0 elif (n==0): return 1 else: return (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b) flatten([[T(n,k,4,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022
Array t(n,k) begins as:
1, 2, 4, 8, 16, ... A000079;
2, 28, 244, 1844, 13260, ...;
4, 244, 5856, 101620, 1511160, ...;
8, 1844, 101620, 3455080, 91981880, ...;
16, 13260, 1511160, 91981880, 4047202720, ...;
32, 93684, 20663388, 2121603436, 146321752612, ...;
64, 657836, 269011408, 44675623468, 4648698508440, ...;
Triangle T(n,k) begins as:
1;
2, 2;
4, 28, 4;
8, 244, 244, 8;
16, 1844, 5856, 1844, 16;
32, 13260, 101620, 101620, 13260, 32;
64, 93684, 1511160, 3455080, 1511160, 93684, 64;
128, 657836, 20663388, 91981880, 91981880, 20663388, 657836, 128;
t[n_, k_, p_, q_]:= t[n, k, p, q] = If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+q)*t[n-1,k,p,q] + (p*n+q)*t[n,k-1,p,q]]];
T[n_, k_, p_, q_]= t[n-k, k, p, q];
Table[T[n,k,5,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 01 2022 *)
@CachedFunction
def t(n,k,p,q):
if (n<0 or k<0): return 0
elif (n==0 and k==0): return 1
else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
def A257614(n,k): return t(n-k,k,5,2)
flatten([[A257614(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 2022
Triangle begins as:
1;
2, 2;
4, 32, 4;
8, 312, 312, 8;
16, 2656, 8736, 2656, 16;
32, 21664, 175424, 175424, 21664, 32;
64, 174336, 3019200, 7016960, 3019200, 174336, 64;
128, 1397120, 47847552, 218838400, 218838400, 47847552, 1397120, 128;
T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,6,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)
def T(n,k,a,b): # A257610 if (k<0 or k>n): return 0 elif (n==0): return 1 else: return (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b) flatten([[T(n,k,6,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022
1;
2, 2;
4, 36, 4;
8, 388, 388, 8;
16, 3676, 12416, 3676, 16;
32, 33564, 283204, 283204, 33564, 32;
64, 303260, 5538184, 13027384, 5538184, 303260, 64;
128, 2732156, 99831564, 465775352, 465775352, 99831564, 2732156, 128;
T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,7,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)
def T(n,k,a,b): # A257617 if (k<0 or k>n): return 0 elif (n==0): return 1 else: return (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b) flatten([[T(n,k,7,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022
Triangle begins as:
1;
2, 2;
4, 44, 4;
8, 564, 564, 8;
16, 6436, 22560, 6436, 16;
32, 71404, 637844, 637844, 71404, 32;
64, 786948, 15470232, 36994952, 15470232, 786948, 64;
128, 8660012, 346391196, 1660722424, 1660722424, 346391196, 8660012, 128;
T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,9,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)
f(x) = 9*x + 2;
t(n, m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, f(m)*t(n-1,m) + f(n)*t(n,m-1)));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", "); ); print(); ); } \\ Michel Marcus, May 23 2015
def T(n,k,a,b): # A257619 if (k<0 or k>n): return 0 elif (n==0): return 1 else: return (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b) flatten([[T(n,k,9,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022
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