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T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
+10
7
1, 0, 1, 0, 2, 2, 0, 18, 6, 3, 0, 192, 48, 12, 4, 0, 2500, 500, 100, 20, 5, 0, 38880, 6480, 1080, 180, 30, 6, 0, 705894, 100842, 14406, 2058, 294, 42, 7, 0, 14680064, 1835008, 229376, 28672, 3584, 448, 56, 8, 0, 344373768, 38263752, 4251528, 472392, 52488, 5832, 648, 72, 9
FORMULA
T(0,0) = 1, else T(n,k) = 0 for k<1 or k>n, else T(n,n) = n, else T(n,k) = (n-1)*n^(n-k).
Sum_{k=0..n} k*T(n,k) = A031972(n).
EXAMPLE
T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,1) = 2: [1,2], [2,1].
T(2,2) = 2: [1,1], [2,2].
T(3,1) = 18: [1,1,2], [1,1,3], [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,2,1], [2,2,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3], [3,3,1], [3,3,2].
T(3,2) = 6: [1,2,2], [1,3,3], [2,1,1], [2,3,3], [3,1,1], [3,2,2].
T(3,3) = 3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 2;
0, 18, 6, 3;
0, 192, 48, 12, 4;
0, 2500, 500, 100, 20, 5;
0, 38880, 6480, 1080, 180, 30, 6;
0, 705894, 100842, 14406, 2058, 294, 42, 7;
0, 14680064, 1835008, 229376, 28672, 3584, 448, 56, 8;
MAPLE
T:= (n, k)-> `if`(n=0 and k=0, 1, `if`(k<1 or k>n, 0,
`if`(k=n, n, (n-1)*n^(n-k)))):
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
f[0, 0]=1;
f[n_, k_]:=Which[1<=k<=n-1, n^(n-k)*(n-1), k<1, 0, k==n, n, k>n, 0];
Table[Table[f[n, k], {k, 0, n}], {n, 0, 10}]//Grid (* Geoffrey Critzer, May 19 2014 *)
Triangle read by rows in which row n lists n+1 terms, starting with n^5 and ending with n^6, such that the difference between successive terms is equal to n^5 - n^4.
+10
6
0, 1, 1, 32, 48, 64, 243, 405, 567, 729, 1024, 1792, 2560, 3328, 4096, 3125, 5625, 8125, 10625, 13125, 15625, 7776, 14256, 20736, 27216, 33696, 40176, 46656, 16807, 31213, 45619, 60025, 74431, 88837, 103243, 117649, 32768, 61440, 90112, 118784, 147456
COMMENTS
The first term of row n is A000584(n) and the last term of row n is A001014(n). Row sums give A163275. - Omar E. Pol, Mar 18 2012
EXAMPLE
Triangle begins:
0;
1,1;
32,48,64;
243,405,567,729;
1024,1792,2560,3328,4096;
3125,5625,8125,10625,13125,15625;
7776,14256,20736,27216,33696,40176,46656;
16807,31213,45619,60025,74431,88837,103243,117649;
32768,61440,90112,118784,147456,176128,204800,233472,262144;
59049,111537,164025,216513,269001,321489,373977,426465,478953,531441;
100000,190000,280000,370000,460000,550000,640000,730000,820000,910000,1000000;
MATHEMATICA
rw[n_]:=Range[n^5, n^6, n^5-n^4]; Join[{0, 1}, Flatten[Array[rw, 10]]] (* Harvey P. Dale, Mar 18 2012 *)
0, 0, 64, 1458, 12288, 62500, 233280, 705894, 1835008, 4251528, 9000000, 17715610, 32845824, 57921708, 97883968, 159468750, 251658240, 386201104, 578207808, 846825858, 1216000000, 1715322420, 2380977984, 3256789558, 4395368448, 5859375000, 7722894400, 10072932714
COMMENTS
For n>1 number of 7-digit positive integers in base n.
FORMULA
a(n) = n^6*(n-1) = n^7 - n^6.
G.f.: 2*(32*x^2 + 473*x^3 + 1208*x^4 + 718*x^5 + 88*x^6 + x^7)/(x - 1)^8. - Wesley Ivan Hurt, Aug 03 2014 Recurrence: a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*(n-6)+8*a(n-7)-a(n-8). - Wesley Ivan Hurt, Aug 03 2014 Sum_{n>=2} 1/a(n) = 6 - Sum_{k=2..6} zeta(k). - Amiram Eldar, Jul 05 2020
MATHEMATICA
CoefficientList[Series[2 (32*x^2 + 473*x^3 + 1208*x^4 + 718*x^5 + 88*x^6 + x^7)/(x - 1)^8, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 03 2014 *)
PROG
(PARI) vector(100, n, (n-1)^7 - (n-1)^6) \\ Derek Orr, Aug 03 2014
0, 0, 128, 4374, 49152, 312500, 1399680, 4941258, 14680064, 38263752, 90000000, 194871710, 394149888, 752982204, 1370375552, 2392031250, 4026531840, 6565418768, 10407740544, 16089691302, 24320000000, 36021770820, 52381515648, 74906159834, 105488842752, 146484375000
COMMENTS
For n>1 number of 8-digit positive integers in base n.
FORMULA
a(n) = n^7*(n-1) = n^8 - n^7.
G.f.: -2*x^2*(x^6+183*x^5+2682*x^4+8422*x^3+7197*x^2+1611*x+64) / (x-1)^9. - Colin Barker, Aug 08 2014 Sum_{n>=2} 1/a(n) = 7 - Sum_{k=2..7} zeta(k). - Amiram Eldar, Jul 05 2020
MATHEMATICA
LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {0, 0, 128, 4374, 49152, 312500, 1399680, 4941258, 14680064}, 30] (* Harvey P. Dale, Apr 29 2016 *)
PROG
(PARI) vector(100, n, (n-1)^8 - (n-1)^7) \\ Derek Orr, Aug 03 2014 (PARI) concat([0, 0], Vec(-2*x^2*(x^6+183*x^5+2682*x^4+8422*x^3+7197*x^2+1611*x+64) / (x-1)^9 + O(x^100))) \\ Colin Barker, Aug 08 2014
0, 0, 256, 13122, 196608, 1562500, 8398080, 34588806, 117440512, 344373768, 900000000, 2143588810, 4729798656, 9788768652, 19185257728, 35880468750, 64424509440, 111612119056, 187339329792, 305704134738, 486400000000, 756457187220, 1152393344256, 1722841676182
COMMENTS
For n>1 number of 9-digit positive integers in base n.
LINKS
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
FORMULA
a(n) = n^8*(n-1) = n^9 - n^8.
G.f.: 2*x^2*(x^7+374*x^6+9327*x^5+49780*x^4+78095*x^3+38454*x^2+5281*x+128) / (x-1)^10. - Colin Barker, Aug 08 2014 Sum_{n>=2} 1/a(n) = 8 - Sum_{k=2..8} zeta(k). - Amiram Eldar, Jul 05 2020
PROG
(PARI) vector(100, n, (n-1)^9 - (n-1)^8) \\ Derek Orr, Aug 03 2014
0, 0, 512, 39366, 786432, 7812500, 50388480, 242121642, 939524096, 3099363912, 9000000000, 23579476910, 56757583872, 127253992476, 268593608192, 538207031250, 1030792151040, 1897406023952, 3372107936256, 5808378560022, 9728000000000, 15885600931620, 25352653573632
COMMENTS
For n>1 number of 10-digit positive integers in base n.
LINKS
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
FORMULA
a(n) = n^9*(n-1) = n^10 - n^9.
G.f.: 2*(256*x^2 + 16867*x^3 + 190783*x^4 + 621199*x^5 + 689155*x^6 + 264409*x^7 + 30973*x^8 + 757*x^9 + x^10)/(1 - x)^11. - Wesley Ivan Hurt, Aug 03 2014 Recurrence: a(n) = 11*a(n-1)-55*a(n-2)+165*a(n-3)-330*a(n-4)+462*a(n-5)-462*a(n-6)+330*a(n-7)-165*a(n-8)+55*a(n-9)-11*a(n-10)+a(n-11). - Wesley Ivan Hurt, Aug 03 2014 Sum_{n>=2} 1/a(n) = 9 - Sum_{k=2..9} zeta(k). - Amiram Eldar, Jul 05 2020
MATHEMATICA
CoefficientList[Series[2 (256*x^2 + 16867*x^3 + 190783*x^4 + 621199*x^5 + 689155*x^6 + 264409*x^7 + 30973*x^8 + 757*x^9 + x^10)/(1 - x)^11, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 03 2014 *) LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {0, 0, 512, 39366, 786432, 7812500, 50388480, 242121642, 939524096, 3099363912, 9000000000}, 40] (* Harvey P. Dale, Oct 19 2022 *)
PROG
(PARI) vector(100, n, (n-1)^10 - (n-1)^9) \\ Derek Orr, Aug 03 2014
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