A287698 -id:A287698 - cp's OEIS frontend

A287697 Triangle read by rows, (Sum_{k=0..n} T[n,k]*x^k) / (1-x)^(n+1) are generating functions of the columns of A287698.

1, 0, 1, 0, 1, 7, 0, 1, 52, 163, 0, 1, 341, 4499, 8983, 0, 1, 2246, 98256, 660746, 966751, 0, 1, 15177, 2045282, 35677082, 155729277, 179781181, 0, 1, 104952, 42658239, 1754605504, 17446464519, 55690144728, 53090086057

Offset: 0

Peter Luschny, May 30 2017

			Triangle starts:
0: [1]
1: [0, 1]
2: [0, 1,      7]
3: [0, 1,     52,      163]
4: [0, 1,    341,     4499,       8983]
5: [0, 1,   2246,    98256,     660746,      966751]
6: [0, 1,  15177,  2045282,   35677082,   155729277,   179781181]
7: [0, 1, 104952, 42658239, 1754605504, 17446464519, 55690144728, 53090086057]
...
Let q4(x) = (x + 341*x^2 + 4499*x^3 + 8983*x^4) / (1-x)^5 then the coefficients of the series expansion of q4 are column 4 of A287698.

Cf. A000442, A212856, A287696, A287698.

A287697_row := n -> Delta(A287696_poly(n), n): # Delta defined in A287315.
for n from 0 to 9 do A287697_row(n) od;
A287697_eulerian := (n,x) -> add(A287697_row(n)[k+1]*x^k,k=0..n)/(1-x)^(n+1):
for n from 0 to 4 do A287697_eulerian(n,x) od;

T(n,n) = A212856(n).

Sum_{k=0..n} T(n,k) = A000442(n).

A287696 Triangle read by rows, T(n,k) = (n!)^3 * [x^k] [z^n] hypergeom([], [1, 1], z)^x for n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, -3, 4, 0, 46, -81, 36, 0, -1899, 3916, -2592, 576, 0, 163476, -375375, 305500, -108000, 14400, 0, -25333590, 63002191, -58725000, 26370000, -5832000, 518400, 0, 6412369860, -16976577828, 17470973569, -9168390000, 2636298000, -400075200, 25401600

Offset: 0

Peter Luschny, May 30 2017

The polynomials Sum_{k=0..n} T(n,k) x^k generate the columns of A287698.

			0: [1]
1: [0,         1]
2: [0,        -3,        4]
3: [0,        46,      -81,        36]
4: [0,     -1899,     3916,     -2592,      576]
5: [0,    163476,  -375375,    305500,  -108000,    14400]
6: [0, -25333590, 63002191, -58725000, 26370000, -5832000, 518400]

T(n,n) = A001044(n).

Cf. A212856, A212855, A287314, A287698.

A287696_row := proc(n) local k; hypergeom([],[1,1],z); series(%^x, z=0, n+1):
n!^3*coeff(%, z, n); seq(coeff(%, x, k), k=0..n) end:
for n from 0 to 8 do A287696_row(n) od;
A287696_poly := proc(n) local k, x; hypergeom([],[1,1],z); series(%^x, z=0, n+1):
unapply(n!^3*coeff(%, z, n), x); end:
for n from 0 to 7 do A287696_poly(n) od;

T[n_, k_] := (n!)^3 SeriesCoefficient[HypergeometricPFQ[{}, {1, 1}, z]^x, {x, 0, k}, {z, 0, n}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 13 2017 *)

Sum_{k=0..n} T(n,k) = A000012(n).

Sum_{k=0..n} abs(T(n,k)) = A212856(n) = A212855_row(3).

A287699 a(n) = (n!)^3 * [x^n] hypergeom([], [1, 1], x)^4.

Original entry on oeis.org

1, 4, 52, 1192, 36628, 1297504, 50419096, 2099649808, 92239977748, 4225417349872, 200149545055552, 9743739840316288, 485293084757188504, 24641572224240907264, 1272101807179840322416, 66620238759427147324192, 3532989709864148362611988, 189447449844340069835395984

Offset: 0

Peter Luschny, May 31 2017

nonn

Row 4 of A287698.

A287699_list := proc(len) series(hypergeom([], [1, 1], x)^4, x, len);
seq((n!)^3*coeff(%, x, n), n=0..len-1) end: A287699_list(18);

Table[SeriesCoefficient[HypergeometricPFQ[{},{1,1},x]^4, {x,0,n}] n!^3, {n,0,17}]

Recurrence: (n-1)^2*n^6*(4179357*n^8 - 78014664*n^7 + 632906379*n^6 - 2915042445*n^5 + 8338804227*n^4 - 15175362645*n^3 + 17163198021*n^2 - 11033807142*n + 3088189672)*a(n) = (n-1)^2*(321810489*n^14 - 6972560595*n^13 + 68195663613*n^12 - 398723185476*n^11 + 1556210780586*n^10 - 4290176939202*n^9 + 8625424194708*n^8 - 12878626068195*n^7 + 14401770594884*n^6 - 12054703508348*n^5 + 7464118767536*n^4 - 3324818312080*n^3 + 1009161337280*n^2 - 187078472000*n + 15996099840)*a(n-1) - (4292199639*n^16 - 114458657040*n^15 + 1414085794758*n^14 - 10748138802219*n^13 + 56276210317086*n^12 - 215315257854096*n^11 + 622952233195566*n^10 - 1390775235633963*n^9 + 2422273491685843*n^8 - 3303218468438722*n^7 + 3516261474581476*n^6 - 2891864865024712*n^5 + 1801769964078784*n^4 - 822311126185120*n^3 + 259313519840640*n^2 - 50491775044480*n + 4574054714880)*a(n-2) + (n-2)^2*(41455042083*n^14 - 1064012746797*n^13 + 12496613326470*n^12 - 88977496857795*n^11 + 428877145565253*n^10 - 1479691251840690*n^9 + 3766745840954286*n^8 - 7184451834695931*n^7 + 10314626748833734*n^6 - 11091478948669399*n^5 + 8794647685753046*n^4 - 4988208446505900*n^3 + 1914324172568200*n^2 - 445291128023840*n + 47406419692800)*a(n-3) + 8*(n-3)^3*(n-2)^2*(87060185667*n^11 - 1712183651451*n^10 + 14956429421367*n^9 - 76542540324894*n^8 + 254862421705026*n^7 - 579589444734966*n^6 + 918686062982112*n^5 - 1015496680681101*n^4 + 767926983449420*n^3 - 378895216669900*n^2 + 109953142874960*n - 14239574392960)*a(n-4) - 175616*(n-4)^3*(n-3)^3*(n-2)^2*(4179357*n^8 - 44579808*n^7 + 203825727*n^6 - 521868123*n^5 + 819229437*n^4 - 808911855*n^3 + 491820735*n^2 - 168723870*n + 25050760)*a(n-5). - Vaclav Kotesovec, Jul 05 2018

a(n) ~ 2^(6*n + 5) / (3^(3/2) * Pi^3 * n^3). - Vaclav Kotesovec, Jul 05 2018

A287700 a(n) = (4!)^3 * [z^4] hypergeom([], [1,1], z)^n.

Original entry on oeis.org

0, 1, 346, 6219, 36628, 124405, 316206, 672511, 1267624, 2189673, 3540610, 5436211, 8006076, 11393629, 15756118, 21264615, 28104016, 36473041, 46584234, 58663963, 72952420, 89703621, 109185406, 131679439, 157481208, 186900025, 220259026, 257895171, 300159244

Offset: 0

Peter Luschny, May 31 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Column 4 of A287698.

[-1899*n + 3916*n^2 - 2592*n^3 + 576*n^4: n in [0..30]]; // Vincenzo Librandi, Jul 20 2017

a := n -> -1899*n + 3916*n^2 - 2592*n^3 + 576*n^4: seq(a(n), n=0..27);

Table[-1899 n + 3916 n^2 - 2592 n^3 + 576 n^4, {n, 0, 30}] (* Bruno Berselli, Jun 06 2017 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 346, 6219, 36628}, 30] (* Vincenzo Librandi, Jul 20 2017 *)

O.g.f.: x*(1 + 341*x + 4499*x^2 + 8983*x^3)/(1 - x)^5.
a(n) = -1899*n + 3916*n^2 - 2592*n^3 + 576*n^4.
a(n) = [x^n] (x + 341*x^2 + 4499*x^3 + 8983*x^4) / (1 - x)^5.

A287701 a(n) = (5!)^3 * [z^5] hypergeom([], [1,1], z)^n.

Original entry on oeis.org

0, 1, 2252, 111753, 1297504, 7120505, 25461756, 70250257, 163191008, 335493009, 629597260, 1100904761, 1819504512, 2871901513, 4362744764, 6416555265, 9179454016, 12820890017, 17535368268, 23544177769, 31097119520, 40474234521, 51987531772, 65982716273, 82840917024, 102980415025, 126858371276, 154972554777

Offset: 0

Peter Luschny, Jun 01 2017

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Column 5 of A287698.

a := n -> 163476*n - 375375*n^2 + 305500*n^3 - 108000*n^4 + 14400*n^5:
seq(a(n), n=0..27);

Table[163476 n - 375375 n^2 + 305500 n^3 - 108000 n^4 + 14400 n^5, {n, 0, 30}] (* Bruno Berselli, Jun 06 2017 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,2252,111753,1297504,7120505},30] (* Harvey P. Dale, Mar 02 2025 *)

O.g.f.: x*(1 + 2246*x + 98256*x^2 + 660746*x^3 + 966751*x^4) / (1 - x)^6.
a(n) = 163476*n - 375375*n^2 + 305500*n^3 - 108000*n^4 + 14400*n^5.
a(n) = [x^n] (x + 2246*x^2 + 98256*x^3 + 660746*x^4 + 966751*x^5) / (1 - x)^6.

A287702 a(n) = (3!)^3 * [z^3] hypergeom([], [1,1], z)^n.

Original entry on oeis.org

0, 1, 56, 381, 1192, 2705, 5136, 8701, 13616, 20097, 28360, 38621, 51096, 66001, 83552, 103965, 127456, 154241, 184536, 218557, 256520, 298641, 345136, 396221, 452112, 513025, 579176, 650781, 728056, 811217, 900480, 996061, 1098176, 1207041, 1322872, 1445885

Offset: 0

Peter Luschny, Jun 01 2017

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Column 3 of A287698.

a := n -> 46*n - 81*n^2 + 36*n^3: seq(a(n), n=0..35);

Table[46 n - 81 n^2 + 36 n^3, {n, 0, 40}] (* Bruno Berselli, Jun 06 2017 *)
LinearRecurrence[{4,-6,4,-1},{0,1,56,381},40] (* Harvey P. Dale, Aug 20 2017 *)

O.g.f.: x*(1 + 52*x + 163*x^2) / (1 - x)^4.
a(n) = 46*n - 81*n^2 + 36*n^3.
a(n) = [x^n] (x + 52*x^2 + 163*x^3) / (1 - x)^4.

cp's OEIS Frontend

A287697 Triangle read by rows, (Sum_{k=0..n} T[n,k]*x^k) / (1-x)^(n+1) are generating functions of the columns of A287698.

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A287696 Triangle read by rows, T(n,k) = (n!)^3 * [x^k] [z^n] hypergeom([], [1, 1], z)^x for n>=0, 0<=k<=n.

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A287699 a(n) = (n!)^3 * [x^n] hypergeom([], [1, 1], x)^4.

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A287700 a(n) = (4!)^3 * [z^4] hypergeom([], [1,1], z)^n.

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A287701 a(n) = (5!)^3 * [z^5] hypergeom([], [1,1], z)^n.

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A287702 a(n) = (3!)^3 * [z^3] hypergeom([], [1,1], z)^n.

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