A103881 -id:A103881 - cp's OEIS frontend

A157074 Number of integer sequences of length n+1 with sum zero and sum of absolute values 50.

2, 150, 6252, 182500, 4112502, 75578370, 1173777752, 15795816120, 187652162502, 1996568642530, 19245807386652, 169668375420180, 1378768046330402, 10396793993805030, 73166155146412752, 482928212647720720, 3002693915693248002, 17655197338344400470

Offset: 1

R. H. Hardin, Feb 22 2009

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (51,-1275,20825,-249900,2349060,-18009460, 115775100,-636763050,3042312350,-12777711870,47626016970,-158753389900, 476260169700,-1292706174900,3188675231420,-7174519270695,14771069086725, -27900908274925,48459472266975,-77535155627160,114456658306760, -156077261327400,196793068630200,-229591913401900,247959266474052, -247959266474052,229591913401900,-196793068630200,156077261327400, -114456658306760,77535155627160,-48459472266975,27900908274925,-14771069086725, 7174519270695,-3188675231420,1292706174900,-476260169700,158753389900, -47626016970,12777711870,-3042312350,636763050,-115775100,18009460,-2349060, 249900,-20825,1275,-51,1).

Cf. A103881, A156554.

A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];
A157074[n_]:= A103881[n, 25];
Table[A157074[n], {n, 50}] (* G. C. Greubel, Jan 27 2022 *)

def A103881(n,k): return sum( binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k) for i in (0..n) )
def A157074(n): return A103881(n, 25)
[A157074(n) for n in (1..50)] # G. C. Greubel, Jan 27 2022

a(n) = T(n,25); T(n,k) = Sum_{i=1..n} binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k).
From G. C. Greubel, Jan 27 2022: (Start)
a(n) = (n+1)*binomial(n+24, 25)*Hypergeometric3F2([-24, -n, 1-n], [2, -n-24], 1).
a(n) = (126410606437752/50!)*n*(n+1)*(9623905480333281923493425053824177930240000000000 + 27100515339271296805042905104567762524569600000000*n + 49226599934719560481828455826236675352166400000000*n^2 + 51923175705445481350794593882064923048017920000000*n^3 + 46502829595021715716879102923565907828539392000000*n^4 + 28607394119885617552139740430561122618473185280000*n^5 + 16559588497213417883781098164439679738807582720000*n^6 + 6903192311627666498917104674104501458397298688000*n^7 + 2894036204442771597885580471785456670461945446400*n^8 + 882529358789488763775646630321568918645729918976*n^9 + 285704714285545970609012303782721701384304001024*n^10 + 66744193695557588078616189319402098781536485376*n^11 + 17394219679949413313652735722550417627568410624*n^12 + 3209212575849629078911083109861120504852463616*n^13 + 693340015644326307061765976396831207893776384*n^14 + 103183723405307213352941409893689330849622016*n^15 + 18890270959451165193941203482138711306505984*n^16 + 2301923694341735297363581288294981193895936*n^17 + 363246399568340082151669298560235347864064*n^18 + 36632957463825141955678003229613126558336*n^19 + 5051271387716061681982819535517710183664*n^20 + 424699960734096109443243714664325553216*n^21 + 51748891662219811557282274201341501784*n^22 + 3644289612230496197746802122023398616*n^23 + 396093870596357042648294916274601009*n^24 + 23416970176809393473086005534732576*n^25 + 2288479608700865971390942858179924*n^26 + 113584302206510356395975946196976*n^27 + 10049618631034902174836327665474*n^28 + 417815336521106587249172637024*n^29 + 33668912037122043295220280476*n^30 + 1166960621063436872100315624*n^31 + 86099204270791153803452751*n^32 + 2468552637980851499947584*n^33 + 167536461123588897837416*n^34 + 3927535896285273089184*n^35 + 246218365513296690316*n^36 + 4640273089678232064*n^37 + 269714108783157936*n^38 + 3986042964314664*n^39 + 215542329647711*n^40 + 2405227111584*n^41 + 121370670916*n^42 + 961331184*n^43 + 45396066*n^44 + 227424*n^45 + 10076*n^46 + 24*n^47 + n^48).
G.f.: 2*x*(1 + 24*x + 576*x^2 + 6624*x^3 + 76176*x^4 + 558624*x^5 + 4096576*x^6 + 21507024*x^7 + 112911876*x^8 + 451647504*x^9 + 1806590016*x^10 + 5720868384*x^11 + 18116083216*x^12 + 46584213984*x^13 + 119787978816*x^14 + 254549454984*x^15 + 540917591841*x^16 + 961631274384*x^17 + 1709566710016*x^18 + 2564350065024*x^19 + 3846525097536*x^20 + 4895577396864*x^21 + 6230734868736*x^22 + 6749962774464*x^23 + 7312459672336*x^24 + 6749962774464*x^25 + 6230734868736*x^26 + 4895577396864*x^27 + 3846525097536*x^28 + 2564350065024*x^29 + 1709566710016*x^30 + 961631274384*x^31 + 540917591841*x^32 + 254549454984*x^33 + 119787978816*x^34 + 46584213984*x^35 + 18116083216*x^36 + 5720868384*x^37 + 1806590016*x^38 + 451647504*x^39 + 112911876*x^40 + 21507024*x^41 + 4096576*x^42 + 558624*x^43 + 76176*x^44 + 6624*x^45 + 576*x^46 + 24*x^47 + x^48)/(1-x)^51. (End)

A103883 Square array A(n,k) read by antidiagonals: coordination sequence for lattice B_n.

Original entry on oeis.org

1, 1, 8, 1, 18, 16, 1, 32, 74, 24, 1, 50, 224, 170, 32, 1, 72, 530, 768, 306, 40, 1, 98, 1072, 2562, 1856, 482, 48, 1, 128, 1946, 6968, 8130, 3680, 698, 56, 1, 162, 3264, 16394, 28320, 20082, 6432, 954, 64, 1, 200, 5154, 34624, 83442, 85992, 42130, 10304, 1250, 72

Offset: 2

Ralf Stephan, Feb 20 2005

			Array, A(n, k), begins:
  1,   8,    16,     24,      32,       40,        48, ... A022144;
  1,  18,    74,    170,     306,      482,       698, ... A022145;
  1,  32,   224,    768,    1856,     3680,      6432, ... A022146;
  1,  50,   530,   2562,    8130,    20082,     42130, ... A022147;
  1,  72,  1072,   6968,   28320,    85992,    214864, ... A022148;
  1,  98,  1946,  16394,   83442,   307314,    907018, ... A022149;
  1, 128,  3264,  34624,  216448,   954880,   3301952, ... A022150;
  1, 162,  5154,  67266,  507906,  2653346,  10666146, ... A022151;
  1, 200,  7760, 122264, 1099040,  6728168,  31208560, ... A022152;
  1, 242, 11242, 210474, 2224178, 15804866,  83999962, ... A022153;
  1, 288, 15776, 346304, 4254912, 34792672, 210482016, ... A022154;
  ...
Antidiagonals, T(n, k), begin as:
  1;
  1,   8;
  1,  18,   16;
  1,  32,   74,    24;
  1,  50,  224,   170,    32;
  1,  72,  530,   768,   306,    40;
  1,  98, 1072,  2562,  1856,   482,   48;
  1, 128, 1946,  6968,  8130,  3680,  698,  56;
  1, 162, 3264, 16394, 28320, 20082, 6432, 954, 64;

G. C. Greubel, Antidiagonals n = 2..50, flattened
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Rows include A022144, A022145, A022146, A022147, A022148, A022149, A022150, A022151, A022152, A022153, A022154.
Columns include A001105.
Cf. A103881, A103884.

A103883:= func< n,k | (&+[Binomial(n-j-1,n-k-1)*(Binomial(2*n-2*k+1,2*j) - 2*j*Binomial(n-k,j)) : j in [0..k]]) >;
[A103883(n,k): k in [0..n-2], n in [2..14]]; // G. C. Greubel, May 24 2023

offset = 2;
T[n_, k_] := SeriesCoefficient[Sum[(Binomial[2n + 1, 2i] - 2i Binomial[n, i]) x^i, {i, 0, n}]/(1 - x)^n, {x, 0, k}];
Table[T[n - k, k], {n, offset, 11}, {k, 0, n - offset}] // Flatten (* Jean-François Alcover, Feb 13 2019 *)

def A103883(n,k): return sum(binomial(n-j-1,n-k-1)*(binomial(2*n-2*k+1,2*j) - 2*j*binomial(n-k,j)) for j in range(k+1))
flatten([[A103883(n,k) for k in range(n-1)] for n in range(2,15)]) # G. C. Greubel, May 24 2023

G.f. of n-th row: (Sum_{i=0..n} (C(2n+1, 2*i) - 2*i*C(n, i))*x^i)/(1-x)^n.
From G. C. Greubel, May 24 2023: (Start)
A(n, k) = Sum_{j=0..k} binomial(n+k-j-1, n-1)*(binomial(2*n+1, 2*j) - 2*j*binomial(n, j)) (array).
T(n, k) = Sum_{j=0..k} binomial(n-j-1, n-k-1)*(binomial(2*n-2*k+1, 2*j) - 2*j*binomial(n-k, j)) (antidiagonals). (End)

A157053 Number of integer sequences of length n+1 with sum zero and sum of absolute values 8.

Original entry on oeis.org

2, 24, 162, 780, 2970, 9492, 26474, 66222, 151560, 322190, 643632, 1219374, 2206932, 3838590, 6447660, 10501172, 16639974, 25727292, 38906870, 57671880, 83945862, 120177024, 169447302, 235597650, 323371100, 438575202, 588265524, 780951962, 1026829680

Offset: 1

R. H. Hardin, Feb 22 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A103881, A156554.

Table[n*(n+1)*(n^2+n+6)*(n^4 +2*n^3 +23*n^2 +22*n +24)/576, {n,50}] (* G. C. Greubel, Jan 23 2022 *)

[n*(n+1)*(n^2+n+6)*(n^4 +2*n^3 +23*n^2 +22*n +24)/576 for n in (1..50)] # G. C. Greubel, Jan 23 2022

a(n) = T(n,4); T(n,k) = Sum_{i=1..n} binomial(n+1,i)*binomial(k-1,i-1)*binomial(n-i+k,k).
G.f.: 2*x*(1+3*x+9*x^2+9*x^3+9*x^4+3*x^5+x^6)/(1-x)^9. - Colin Barker, Mar 17 2012
a(n) = n*(n+1)*(n^2+n+6)*(n^4 +2*n^3 +23*n^2 +22*n +24)/576. - Bruno Berselli, Mar 17 2012
E.g.f.: (x/576)*(1152 +5760*x +9216*x^2 +6432*x^3 +2208*x^4 +384*x^5 +32*x^6 +x^7)*exp(x). - G. C. Greubel, Jan 23 2022

A157054 Number of integer sequences of length n+1 with sum zero and sum of absolute values 10.

Original entry on oeis.org

2, 30, 252, 1500, 7002, 27174, 91112, 271224, 731502, 1815506, 4197468, 9129276, 18827718, 37060506, 70006512, 127485584, 224676522, 384468534, 640622012, 1041949020, 1657762722, 2584888350, 3956576472, 5953712520, 8818775030, 12873059082, 18537751260

Offset: 1

R. H. Hardin, Feb 22 2009

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A103881, A156554.

Table[n*(n+1)*(n^8 +4*n^7 +66*n^6 +184*n^5 +1089*n^4 +1876*n^3 +4604*n^2 +3696*n +2880)/14400, {n,50}] (* G. C. Greubel, Jan 23 2022 *)

[n*(n+1)*(n^8 +4*n^7 +66*n^6 +184*n^5 +1089*n^4 +1876*n^3 +4604*n^2 +3696*n +2880)/14400 for n in (1..50)] # G. C. Greubel, Jan 23 2022

a(n) = T(n,5); T(n,k) = Sum_{i=1..n} binomial(n+1,i)*binomial(k-1,i-1)*binomial(n-i+k,k).
G.f.: 2*x*(1+4*x+16*x^2+24*x^3+36*x^4+24*x^5+16*x^6+4*x^7+x^8)/(1-x)^11. - Colin Barker, Mar 17 2012
From G. C. Greubel, Jan 23 2022: (Start)
a(n) = n*(n+1)*(n^8 +4*n^7 +66*n^6 +184*n^5 +1089*n^4 +1876*n^3 +4604*n^2 +3696*n +2880)/14400.
E.g.f.: (x/14400)*(28800 +187200*x +403200*x^2 +398400*x^3 +207840*x^4 +61200*x^5 +10400*x^6 +1000*x^7 +50*x^8 +x^9)*exp(x). (End)

A157055 Number of integer sequences of length n+1 with sum zero and sum of absolute values 12.

Original entry on oeis.org

2, 36, 362, 2570, 14240, 65226, 256508, 889716, 2777370, 7925720, 20934474, 51697802, 120353324, 265953170, 561075720, 1135620536, 2214405618, 4175000796, 7634582090, 13577591370, 23539760552, 39868752506, 66087441092, 107392877100, 171332460650, 268708978512

Offset: 1

R. H. Hardin, Feb 22 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Column k=6 of A103881.

Cf. A156554.

Table[n*(n+1)*(n^10 +5*n^9 +120*n^8 +450*n^7 +4173*n^6 +10965*n^5 +48530*n^4 +79300*n^3 +163176*n^2 +125280*n +86400)/518400, {n, 50}] (* G. C. Greubel, Jan 24 2022 *)

[n*(n+1)*(n^10 +5*n^9 +120*n^8 +450*n^7 +4173*n^6 +10965*n^5 +48530*n^4 +79300*n^3 +163176*n^2 +125280*n +86400)/518400 for n in (1..50)] # G. C. Greubel, Jan 24 2022

a(n) = T(n,6) where T(n,k) = Sum_{i=1..n} binomial(n+1,i)*binomial(k-1,i-1)*binomial(n-i+k,k).
G.f.: 2*x*(1 +5*x +25*x^2 +50*x^3 +100*x^4 +100*x^5 +100*x^6 +50*x^7 +25*x^8 +5*x^9 +x^10)/(1-x)^13. - Colin Barker, Jan 25 2013
From G. C. Greubel, Jan 24 2022: (Start)
a(n) = n*(n+1)*(n^10 +5*n^9 +120*n^8 +450*n^7 +4173*n^6 +10965*n^5 +48530*n^4 +79300*n^3 +163176*n^2 +125280*n +86400)/518400.
E.g.f.: (x/518400)*(1036800 +8294400*x +22464000*x^2 +28728000*x^3 +20131200*x^4 +8369280*x^5 +2154240*x^6 +349200*x^7 +35400*x^8 +2160*x^9 +72*x^10 +x^11)*exp(x). (End)

A157056 Number of integer sequences of length n+1 with sum zero and sum of absolute values 14.

Original entry on oeis.org

2, 42, 492, 4060, 26070, 137886, 623576, 2476296, 8809110, 28512110, 85014204, 235895244, 614266354, 1511679210, 3536846160, 7907476016, 16967926746, 35078339106, 70098276620, 135798494460, 255689552382, 468969729382, 839584669992, 1469778991800, 2520031983950

Offset: 1

R. H. Hardin, Feb 22 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. A103881, A156554.

Table[n*(n+1)*(n^12 +6*n^11 +197*n^10 +930*n^9 +12363*n^8 +43938*n^7 +300551*n^6 +751710*n^5 +2756536*n^4 +4309656*n^3 +7816752*n^2 +5780160*n +3628800)/25401600, {n,50}] (* G. C. Greubel, Jan 24 2022 *)

[n*(n+1)*(n^12 +6*n^11 +197*n^10 +930*n^9 +12363*n^8 +43938*n^7 +300551*n^6 +751710*n^5 +2756536*n^4 +4309656*n^3 +7816752*n^2 +5780160*n +3628800)/25401600 for n in (1..50)] # G. C. Greubel, Jan 24 2022

a(n) = T(n,7); T(n,k) = Sum_{i=1..n} binomial(n+1,i)*binomial(k-1,i-1)*binomial(n-i+k,k).
G.f.: 2*x*(1 +6*x +36*x^2 +90*x^3 +225*x^4 +300*x^5 +400*x^6 +300*x^7 +225*x^8 +90*x^9 +36*x^10 +6*x^11 +x^12)/(1-x)^15. - Colin Barker, Jan 25 2013
From G. C. Greubel, Jan 24 2022: (Start)
a(n) = n*(n+1)*(n^12 +6*n^11 +197*n^10 +930*n^9 +12363*n^8 +43938*n^7 +300551*n^6 +751710*n^5 +2756536*n^4 +4309656*n^3 +7816752*n^2 +5780160*n +3628800)/25401600.
E.g.f.: (x/25401600)*(50803200 +482630400*x +1574899200*x^2 +2472422400*x^3 +2176070400*x^4 +1169320320*x^5 +403683840*x^6 +92221920*x^7 +14129640*x^8 +1449420*x^9 +97608*x^10 +4116*x^11 +98*x^12 +x^13)*exp(x). (End)

A157057 Number of integer sequences of length n+1 with sum zero and sum of absolute values 16.

Original entry on oeis.org

2, 48, 642, 6040, 44130, 264936, 1356194, 6077196, 24314490, 88206140, 293744154, 907129236, 2619716554, 7125357540, 18363363690, 45076309166, 105864434424, 238815143406, 519252051080, 1091481669390, 2224042468032, 4403475647758, 8489857618992, 15969368635950

Offset: 1

R. H. Hardin, Feb 22 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Cf. A103881, A156554.

Table[(12870/16!)*n*(n+1)*(203212800 +349090560*n +487608192*n^2 +296058000*n^3 +196660016*n^4 +61391512*n^5 +25601072*n^6 +4564385*n^7 + 1344383*n^8 + 138621*n^9 +30835*n^10 +1715*n^11 +301*n^12 +7*n^13 +n^14), {n, 50}] (* G. C. Greubel, Jan 24 2022 *)

[(12870/factorial(16))*n*(n+1)*(203212800 +349090560*n +487608192*n^2 +296058000*n^3 +196660016*n^4 +61391512*n^5 +25601072*n^6 +4564385*n^7 + 1344383*n^8 + 138621*n^9 +30835*n^10 +1715*n^11 +301*n^12 +7*n^13 +n^14) for n in (1..50)] # G. C. Greubel, Jan 24 2022

a(n) = T(n,8); T(n,k) = Sum_{i=1..n} binomial(n+1,i)*binomial(k-1,i-1)*binomial(n-i+k,k).
G.f.: 2*x*(1 + 7*x + 49*x^2 + 147*x^3 + 441*x^4 + 735*x^5 + 1225*x^6 + 1225*x^7 + 1225*x^8 + 735*x^9 + 441*x^10 + 147*x^11 + 49*x^12 + 7*x^13 + x^14)/(1-x)^17. - Colin Barker, Jan 25 2013
a(n) = (12870/16!)*n*(n+1)*(203212800 + 349090560*n + 487608192*n^2 + 296058000*n^3 + 196660016*n^4 + 61391512*n^5 + 25601072*n^6 + 4564385*n^7 + 1344383*n^8 + 138621*n^9 + 30835*n^10 + 1715*n^11 + 301*n^12 + 7*n^13 + n^14). - G. C. Greubel, Jan 24 2022

A157058 Number of integer sequences of length n+1 with sum zero and sum of absolute values 18.

Original entry on oeis.org

2, 54, 812, 8580, 70310, 472626, 2703512, 13507416, 60110030, 241925530, 891454124, 3037849828, 9654482474, 28818500830, 81289041680, 217815522736, 556959705302, 1364497268946, 3214138597460, 7302195414780, 16045139112002, 34183012888134, 70764981877592

Offset: 1

R. H. Hardin, Feb 22 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (19,-171,969,-3876,11628,-27132,50388,-75582, 92378,-92378,75582,-50388,27132,-11628,3876,-969,171,-19,1).

Cf. A103881, A156554.

Table[(48620/18!)*n*(n+1)*(14631321600 +26760222720*n +38452817664*n^2 +25217041536*n^3 +17311651344*n^4 +5993468992*n^5 +2592460808*n^6 +533444296*n^7 +163476113*n^8 +20735776*n^9 +4812092*n^10 +370160*n^11 +67942*n^12 +2912*n^13 +436*n^14 +8*n^15 +n^16), {n,50}] (* G. C. Greubel, Jan 24 2022 *)

[(48620/factorial(18))*n*(n+1)*(14631321600 +26760222720*n +38452817664*n^2 +25217041536*n^3 +17311651344*n^4 +5993468992*n^5 +2592460808*n^6 +533444296*n^7 +163476113*n^8 +20735776*n^9 +4812092*n^10 +370160*n^11 +67942*n^12 +2912*n^13 +436*n^14 +8*n^15 +n^16) for n in (1..50)] # G. C. Greubel, Jan 24 2022

a(n) = T(n,9); T(n,k) = Sum_{i=1..n} binomial(n+1,i)*binomial(k-1,i-1)*binomial(n-i+k,k).
G.f.: 2*x*(1 +8*x +64*x^2 +224*x^3 +784*x^4 +1568*x^5 +3136*x^6 +3920*x^7 +4900*x^8 +3920*x^9 +3136*x^10 +1568*x^11 +784*x^12 +224*x^13 +64*x^14 +8*x^15 +x^16)/(1-x)^19. - Colin Barker, Jan 25 2013
a(n) = (48620/18!)*n*(n+1)*(14631321600 +26760222720*n +38452817664*n^2 +25217041536*n^3 +17311651344*n^4 +5993468992*n^5 +2592460808*n^6 +533444296*n^7 +163476113*n^8 +20735776*n^9 +4812092*n^10 +370160*n^11 +67942*n^12 +2912*n^13 +436*n^14 +8*n^15 +n^16). - G. C. Greubel, Jan 24 2022

A157059 Number of integer sequences of length n+1 with sum zero and sum of absolute values 20.

Original entry on oeis.org

2, 60, 1002, 11750, 106752, 794598, 5025692, 27717948, 135916002, 601585512, 2432878866, 9079799742, 31534801116, 102644594262, 315029394792, 916470530808, 2538818182782, 6724224543708, 17088309885542, 41800229045610, 98698280879352, 225524301678170

Offset: 1

R. H. Hardin, Feb 22 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (21,-210,1330,-5985,20349,-54264, 116280,-203490,293930,-352716, 352716,-293930,203490,-116280,54264,-20349,5985, -1330,210,-21,1).

Cf. A103881, A156554.

A103881[n_, k_]:= (n+1)*Binomial[n+k-1, k]*HypergeometricPFQ[{1-n,-n,1-k}, {2,1-n-k}, 1];
A157059[n_]:= A103881[n,10];
Table[A157059[n], {n, 50}] (* G. C. Greubel, Jan 24 2022 *)

def A103881(n,k): return sum( binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k) for i in (0..n) )
def A157059(n): return A103881(n, 10)
[A157059(n,10) for n in (1..50)] # G. C. Greubel, Jan 24 2022

a(n) = T(n,10); T(n,k) = Sum_{i=1..n} binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k).
G.f.: 2*x*(x^18 +9*x^17 +81*x^16 +324*x^15 +1296*x^14 +3024*x^13 +7056*x^12 +10584*x^11 +15876*x^10 +15876*x^9 +15876*x^8 +10584*x^7 +7056*x^6 +3024*x^5 +1296*x^4 +324*x^3 +81*x^2 +9*x +1)/(1-x)^21. - Colin Barker, Jan 25 2013
From G. C. Greubel, Jan 24 2022: (Start)
a(n) = (184756/20!)*n*(n+1)*(1316818944000 +2540101939200*n +3742987138560*n^2 +2615609097216*n^3 +1848671853984*n^4 +695217071376*n^5 +310567813984*n^6 +71342133912*n^7 +22639753938*n^8 +3337504857*n^9 +803922153*n^10 +76623228*n^11 +14628472*n^12 +873558*n^13 +136302*n^14 +4644*n^15 +606*n^16 +9*n^17 +n^18).
a(n) = (n+1)*binomial(n+9, 10)*Hypergeometric3F2([-9, -n, 1-n], [2, -n-9], 1). (End)

A157060 Number of integer sequences of length n+1 with sum zero and sum of absolute values 22.

Original entry on oeis.org

2, 66, 1212, 15620, 155850, 1272810, 8823080, 53265960, 285510150, 1379301990, 6078578508, 24680519604, 93093230958, 328512273390, 1091144804400, 3429182092560, 10244035242630, 29206656395910, 79759293448100

Offset: 1

R. H. Hardin, Feb 22 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (23,-253,1771,-8855,33649,-100947,245157,-490314, 817190,-1144066,1352078, -1352078,1144066,-817190,490314,-245157,100947,-33649, 8855,-1771,253,-23,1).

Cf. A103881, A156554.

A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2,1-n - k}, 1];
A157060[n_] := A103881[n, 11];
Table[A157060[n], {n, 50}] (* G. C. Greubel, Jan 24 2022 *)

def A103881(n,k): return sum( binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k) for i in (0..n) )
def A157060(n): return A103881(n, 11)
[A157060(n) for n in (1..50)] # G. C. Greubel, Jan 24 2022

a(n) = T(n,11); T(n,k) = Sum_{i=1..n} binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k).
From G. C. Greubel, Jan 24 2022: (Start)
a(n) = (n+1)*binomial(n+10, 11)*Hypergeometric3F2([-10, -n, 1-n], [2, -n-10], 1).
a(n) = (705432/22!)*n*(n+1)*(144850083840000 +292579402752000*n +440986525516800*n^2 +325146872079360*n^3 +235868591146176*n^4 +94960596391200*n^5 +43658519177360*n^6 +10953312870160*n^7 +3585704220196*n^8 +593523073650*n^9 +147783744195*n^10 +16467776610*n^11 +3255909581*n^12 +242376100*n^13 +39230830*n^14 +1873860*n^15 +254046*n^16 +7050*n^17 +815*n^18 +10*n^19 +n^20).
G.f.: 2*x*(1 +10*x +100*x^2 +450*x^3 +2025*x^4 +5400*x^5 +14400*x^6 +25200*x^7 +44100*x^8 +52920*x^9 +63504*x^10 +52920*x^11 +44100*x^12 +25200*x^13 +14400*x^14 +5400*x^15 +2025*x^16 +450*x^17 +100*x^18 +10*x^19 +x^20)/(1-x)^23. (End)

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A157074 Number of integer sequences of length n+1 with sum zero and sum of absolute values 50.

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A103883 Square array A(n,k) read by antidiagonals: coordination sequence for lattice B_n.

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A157053 Number of integer sequences of length n+1 with sum zero and sum of absolute values 8.

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A157054 Number of integer sequences of length n+1 with sum zero and sum of absolute values 10.

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A157055 Number of integer sequences of length n+1 with sum zero and sum of absolute values 12.

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A157056 Number of integer sequences of length n+1 with sum zero and sum of absolute values 14.

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A157057 Number of integer sequences of length n+1 with sum zero and sum of absolute values 16.

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A157058 Number of integer sequences of length n+1 with sum zero and sum of absolute values 18.

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