A154283 -id:A154283 - cp's OEIS frontend

A174264 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)(1-x)^(3n+1)Sum_{k >= 0} (k(k+1)(2k+1)/6)^n*x^k and p(0, x) = 1, read by rows.

1, 1, 1, 1, 18, 42, 18, 1, 1, 115, 1539, 5065, 5065, 1539, 115, 1, 1, 612, 30369, 359056, 1439038, 2255448, 1439038, 359056, 30369, 612, 1, 1, 3109, 487944, 16069256, 177275075, 808273143, 1688579472, 1688579472, 808273143, 177275075, 16069256, 487944, 3109, 1

Offset: 0

Roger L. Bagula, Mar 14 2010

			Irregular triangle begins as:
  1;
  1,   1;
  1,  18,    42,     18,       1;
  1, 115,  1539,   5065,    5065,    1539,     115,      1;
  1, 612, 30369, 359056, 1439038, 2255448, 1439038, 359056, 30369, 612, 1;

G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

Cf. A060187, A154283.

(* First program *)
p[n_, x_]:= p[n,x]= If[n==0, 1, (1-x)^(3*n+1)*Sum[(k*(k+1)*(2*k+1)/6)^n*x^k, {k, 0, Infinity}]/x];
Table[CoefficientList[p[n, x], x], {n,0,10}]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k]= If[n<2, Binomial[n, k], Sum[(-1)^(k-j+1)*Binomial[3*n+1, k-j +1]*(j*(1+j)*(1+2*j)/6)^n, {j,0,k+1}]];
Join[{1}, Table[T[n, k], {n,0,10}, {k,0,3*n-2}]//Flatten] (* G. C. Greubel, Mar 25 2022 *)

@CachedFunction
def T(n,k):
    if (n<2): return binomial(n,k)
    else: return sum( (-1)^(k-j+1)*binomial(3*n+1, k-j+1)*(j*(1+j)*(1+2*j)/6)^n for j in (0..k+1) )
[1]+flatten([[T(n,k) for k in (0..3*n-2)] for n in (0..10)]) # G. C. Greubel, Mar 25 2022

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^(3*n+1)*Sum_{k >= 0} (k*(k+1)*(2*k+1)/6)^n*x^k and p(0, x) = 1.

T(n, k) = Sum_{j=0..k+1} (-1)^(k-j+1)*binomial(3*n+1, k-j+1)*(j*(1+j)*(1+2*j)/6)^n, with T(0, k) = T(1, k) = 1. - G. C. Greubel, Mar 25 2022

Edited by G. C. Greubel, Mar 25 2022

A151626 Number of permutations of 2 indistinguishable copies of 1..n with exactly 4 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 0, 1, 603, 47290, 1822014, 49258935, 1086859301, 21147576440, 379269758400, 6441229796061, 105398150289775, 1680774708581766, 26324724399068626, 407112461900381715, 6239897666881158537, 95036195852840662820, 1440959515956284422196, 21778829725476446172249

Offset: 1

R. H. Hardin, May 29 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (70, -2163, 39122, -462914, 3792972, -22214806, 94629124, -295393077, 675442494, -1122134391, 1331169066, -1093400856, 588623760, -186332400, 26244000).

Column k=4 of A154283.

A151626:= func< n | (&+[(-1)^(j+1)*Binomial(2*n+1,5-j)*Binomial(j+1,2)^n: j in [1..5]]) >;
[A151626(n): n in [1..30]]; // G. C. Greubel, Sep 07 2022

With[{B=Binomial}, Table[Sum[(-1)^(j+1)*B[2n+1, 5-j]*B[j+1,2]^n, {j,5}], {n,30}]] (* G. C. Greubel, Sep 07 2022 *)

a(n) = {15^n - (2*n + 1)*10^n + binomial(2*n+1, 2)*6^n - binomial(2*n+1, 3)*3^n + binomial(2*n+1, 4) } \\ Andrew Howroyd, May 07 2020

@CachedFunction
def A151626(n): return sum((-1)^(j+1)*binomial(2*n+1,5-j)*binomial(j+1,2)^n for j in (1..5))
[A151626(n) for n in (1..30)] # G. C. Greubel, Sep 07 2022

a(n) = 15^n - (2*n + 1)*10^n + binomial(2*n+1, 2)*6^n - binomial(2*n+1, 3)*3^n + binomial(2*n+1, 4). - Andrew Howroyd, May 07 2020
From G. C. Greubel, Sep 07 2022: (Start)
G.f.: x^3*(1 + 533*x + 7243*x^2 - 223119*x^3 + 878573*x^4 + 5014923*x^5 - 40074183*x^6 + 75062403*x^7 + 19732086*x^8 - 185394420*x^9 + 117543960*x^10 + 43740000*x^11)/((1-x)^5*(1-3*x)^4*(1-6*x)^3*(1-10*x)^2*(1-15*x)).
E.g.f.: exp(15*x) - (1+20*x)*exp(10*x) + 18*x*(1+4*x)*exp(6*x) - 3*x*(1 + 12*x + 12*x^2)*exp(3*x) + (x^2/6)*(15 + 20*x + 4*x^2)*exp(x). (End)

Terms a(12) and beyond from Andrew Howroyd, May 07 2020

A151627 Number of permutations of 2 indistinguishable copies of 1..n with exactly 5 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 0, 0, 72, 27664, 2864328, 163809288, 6727188848, 225167210712, 6590156148912, 175992170793456, 4407169187423736, 105396936343707456, 2437638848729751736, 55010494951127561400, 1219075824289276443744, 26652917330108137129544, 576864003740129587504224

Offset: 1

R. H. Hardin, May 29 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (126, -7245, 252750, -5998905, 102950910, -1325335665, 13104159030, -101064788055, 614053303570, -2956203883287, 11303748373482, -34308033838515, 82354980144330, -155288585585115, 227560437228978, -255076375732668, 213622511296680, -128922602864400, 52813189404000, -13115963880000, 1488034800000).

Column k=5 of A154283.

[(&+[(-1)^j*Binomial(2*n+1, j)*Binomial(7-j, 2)^n: j in [0..5]]): n in [1..30]]; // G. C. Greubel, Sep 07 2022

With[{B = Binomial}, Table[Sum[(-1)^j*B[2n+1,j]*B[7-j,2]^n, {j,0,5}], {n, 30}]] (* G. C. Greubel, Sep 07 2022 *)

@CachedFunction
def A151627(n): return sum((-1)^j*binomial(2*n+1, j)*binomial(7-j, 2)^n for j in (0..5))
[A151627(n) for n in (1..30)] # G. C. Greubel, Sep 07 2022

From G. C. Greubel, Sep 07 2022: (Start)
a(n) = Sum_{j=0..5} (-1)^j*binomial(2*n+1, j)*binomial(7-j, 2)^n.
G.f.: 8*x^4*(9 + 2324*x - 12462*x^2 - 1858545*x^3 + 34890010*x^4 - 134744022*x^5 - 1875623070*x^6 + 22965673068*x^7 - 95590873845*x^8 + 93562460910*x^9 + 576877450068*x^10 - 2203266593259*x^11 + 2865061552194*x^12 - 347005909980*x^13 - 2472141497400*x^14 + 1471264884000*x^15 + 318864600000*x^16)/((1-x)^6*(1-3*x)^5*(1-6*x)^4*(1-10*x)^3*(1-15*x)^2*(1-21*x)).
E.g.f.: exp(21*x) - (1 + 30*x)*exp(15*x) + 10*x*(3 + 20*x)*exp(10*x) - 6*x*(1 + 24*x + 48*x^2)*exp(6*x) + (9*x^2/2)*(5 + 20*x + 12*x^2)*exp(3*x) - (x^2/30)*(15 + 90*x + 60*x^2 + 8*x^3)*exp(x). (End)

Terms a(11) and beyond from Andrew Howroyd, May 06 2020

A151628 Number of permutations of 2 indistinguishable copies of 1..n with exactly 6 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 0, 0, 1, 5158, 1822014, 242384856, 19323413187, 1130781824398, 54076536713976, 2251621794635088, 84973986733001061, 2985450779006443846, 99474230412387811666, 3185003930126491696920, 98939258210106714816135, 3003063241991742340646382, 89537653738976723063722828

Offset: 1

R. H. Hardin, May 29 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (210, -20559, 1249006, -52877484, 1660698792, -40217937324, 770684131800, -11898983656350, 149952459677980, -1556983686224082, 13409725967210820, -96243862494272068, 577309836510214632, -2898323263572570108, 12179135493109203192, -42783931230910840233, 125321632824303394722, -304873695791112375063, 612528671944484732862, -1008704075532213688776, 1347947980011397405152, -1442243815707288575520, 1213235943921688622400, -782048981957040864000, 371788922872056960000, -122537228378997600000, 24957111340728000000, -2362404048480000000).

Column k=6 of A154283.

[(&+[(-1)^j*Binomial(2*n+1, j)*Binomial(8-j, 2)^n: j in [0..6]]): n in [1..30]]; // G. C. Greubel, Sep 07 2022

With[{B=Binomial}, Table[Sum[(-1)^j*B[2n+1,j]*B[8-j,2]^n, {j,0,6}], {n, 30}]] (* G. C. Greubel, Sep 07 2022 *)

@CachedFunction
def A151628(n): return sum((-1)^j*binomial(2*n+1, j)*binomial(8-j, 2)^n for j in (0..6))
[A151628(n) for n in (1..30)] # G. C. Greubel, Sep 07 2022

From G. C. Greubel, Sep 07 2022: (Start)
a(n) = Sum_{j=0..6} (-1)^j*binomial(2*n+1, j)*binomial(8-j, 2)^n.
G.f.: x^4*(1 + 4948*x + 759393*x^2 - 35443768*x^3 - 508116211*x^4 + 51430255228*x^5 - 1039884450243*x^6 + 5791934217096*x^7 + 99233948186819*x^8 - 2137209451932636*x^9 + 17699047175646675*x^10 - 64844223652304424*x^11 - 67279992193011969*x^12 + 1850800989665593044*x^13 - 8839633922267140593*x^14 + 20366483030687973816*x^15 - 15348635039953199376*x^16 - 39686222209918929480*x^17 + 123668352881463084480*x^18 - 135232901326862200800*x^19 + 35906630373023328000*x^20 + 48364304383014480000*x^21 - 29287301536936800000*x^22 - 4134207084840000000*x^23)/((1-x)^7*(1-3*x)^6*(1-6*x)^5*(1-10*x)^4*(1-15*x)^3*(1-21*x)^2*(1-28*x)).
E.g.f.: exp(28*x) - (1 + 42*x)*exp(21*x) + 45*x*(1 + 10*x)*exp(15*x) - (10/3)*x*(3 + 120*x + 400*x^2)*exp(10*x) + 18*x^2*(5 + 40*x + 48*x^2)*exp(6*x) - (9/10)*x^2*(5 + 90*x + 180*x^2 + 72*x^3)*exp(3*x) + (1/90)*x^3*(105 + 210*x + 84*x^2 + 8*x^3)*exp(x). (End)

Terms a(11) and beyond from Andrew Howroyd, May 06 2020

A151629 Number of permutations of 2 indistinguishable copies of 1..n with exactly 7 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 0, 0, 0, 232, 450048, 163809288, 27306899520, 2898916824320, 230479103253264, 15045786224718576, 853790829031070016, 43726349865720132216, 2073954076439134340896, 92786533933117718314680, 3968339124661862533557120, 163867469809854783921566544

Offset: 1

R. H. Hardin, May 29 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (330, -51513, 5066402, -352805739, 18532822542, -764011192951, 25389927552654, -693360910567062, 15782229920923084, -302672805175992858, 4931504254236896916, -68703084605396486102, 822462579151947202524, -8492510968865324313198, 75847875003485374953052, -587051007000593357006397, 3942243867122711876178882, -22980348438736769272554525, 116263149690925646738764650, -510093882106104569146940943, 1937990696541806422436512950, -6362481040931909732744369259, 17998658175922136342871558966, -43712735312807208911181978972, 90728113301609002223916131208, -160019275941921567466090870848, 238141442344924194341467169088, -296416786921079653107637015680, 305161844579852353104555820800, -256124655172887234493061088000, 171916771042339597743180480000, -89854952889081931534972800000, 35162508668121498752928000000, -9674457989784467806080000000, 1666424129434910092800000000, -135019896025206528000000000).

Column k=7 of A154283.

[(&+[(-1)^j*Binomial(2*n+1, j)*Binomial(9-j, 2)^n: j in [0..7]]): n in [1..30]]; // G. C. Greubel, Sep 08 2022

With[{B=Binomial}, Table[Sum[(-1)^j*B[2n+1, j]*B[9-j, 2]^n, {j,0,7}], {n, 30}]] (* G. C. Greubel, Sep 08 2022 *)

def A151629(n): return sum((-1)^j*binomial(2*n+1, j)*binomial(9-j, 2)^n for j in (0..7))
[A151629(n) for n in (1..30)] # G. C. Greubel, Sep 08 2022

From G. C. Greubel, Sep 07 2022: (Start)
a(n) = Sum_{j=0..7} (-1)^j*binomial(2*n+1, j)*binomial(9-j, 2)^n.
G.f.: 8*x^5*(29 +46686*x +3405558*x^2 -592781020*x^3 +15959334952*x^4 +631633031922*x^5 -49837254287872*x^6 +1300883760100354*x^7 -12994364551718898*x^8 -140176079949572802*x^9 +6513756576348329884*x^10 -101042319163019645166*x^11 +848633388017107293828*x^12 -2913665757033808948194*x^13 -19357175742148303993152*x^14 +332871592406004436180230*x^15 -2265050438781150240585891*x^16 +8844782645551069762176780*x^17 -16577175062101039893470178*x^18 -216035122652452146094327988*x^19 +244246494424905520901547660*x^20 -780226424729404888409973432*x^21 +1345511462530423731597208080*x^22 -1027054667766768116706056160*x^23 -747115159033132605830894400*x^24 +2731966566484322974432464000*x^25 -2760478881311463186555360000*x^26 +892027667079782450985600000*x^27 +450814927116061418400000000*x^28 -303214961231096241600000000*x^29 -30004421338934784000000000*x^30)/( Product_{j=1..8} (1-binomial(j+1,2)*x)^(9-j) ).
E.g.f.: exp(36*x) - (1 + 56*x)*exp(28*x) + 63*x*(1 + 14*x)*exp(21*x) - 15*x*(1 + 60*x + 300*x^2)*exp(15*x) + (250/3)*x^2*(3 + 40*x + 80*x^2)*exp(10*x) - (18/5)*x^2*(5 + 180*x + 720*x^2 + 576*x^3)*exp(6*x) + (9/10)*x^3*(35 + 210*x + 252*x^2 + 72*x^3)*exp(3*x) - (1/630)*x^3*(105 + 840*x + 840*x^2 + 224*x^3 + 16*x^4)*exp(x). (End)

Terms a(11) and beyond from Andrew Howroyd, May 06 2020

A151630 Number of permutations of 2 indistinguishable copies of 1..n with exactly 8 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 0, 0, 0, 1, 37257, 49258935, 19323413187, 3950966047950, 539417838175698, 56160822639510114, 4828612774471173450, 360918591663105680031, 24285778099889122541071, 1507815882167268489272385, 87937588306397361416746005, 4882223035755085016119166100

Offset: 1

R. H. Hardin, May 29 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
G. C. Greubel, Generating functions
Index entries for linear recurrences with constant coefficients, signature (495, -117117, 17647773, -1903962060, 156757247010, -10251395632590, 547263937214190, -24324431827594815, 913604942997437285, -29330261021755017639, 812145644279366953815, -19536467308476604835118, 410659658077014965566332, -7578696975148131792068340, 123267861018665262572785476, -1772533955817200772006505695, 22589412092760272526249081417, -255637202218592552368928158843, 2572704107472364912183081654395, -23049204197252732334976831299504, 183953147111817401024161390520370, -1308189874199616697817539200074238, 8289399765160712341131849367972542, -46783616321998478795458149057122145, 234999225964232933834149112694372867, -1049484407925242792828750467607365185, 4161052733141162438991333881629966449, -14620606470238507069638288669331531446, 45425825709704628283109430191976080400, -124468763777608175568796041188271717984, 299821137334135681782712542449338805040, -632525928565428301410709620694018226208, 1163532979856355964909411340878324232832, -1856399981685264274584742456532993660160, 2552803207326431764967213761527030470400, -3002667039899658352436610420272301120000, 2992862237498353265892271006627584000000, -2498577025975608447805248342475776000000, 1721249910903347055783181973286720000000, -959341910144268687109525373846400000000, 420984649108341364830441158592000000000, -139779856035015771303825062400000000000, 32957940077972879881652858880000000000, -4910962939069964235908198400000000000, 347259290825980971841536000000000000).

Column k=8 of A154283.

[(&+[(-1)^j*Binomial(2*n+1, j)*Binomial(10-j, 2)^n: j in [0..8]]): n in [1..30]]; // G. C. Greubel, Sep 08 2022

With[{B=Binomial}, Table[Sum[(-1)^j*B[2n+1,j]*B[10-j,2]^n, {j,0,8}], {n, 30}]] (* G. C. Greubel, Sep 08 2022 *)

def A151630(n): return sum((-1)^j*binomial(2*n+1, j)*binomial(10-j, 2)^n for j in (0..8))
[A151630(n) for n in (1..30)] # G. C. Greubel, Sep 08 2022

From G. C. Greubel, Sep 08 2022: (Start)
a(n) = Sum_{j=0..8} (-1)^j*binomial(2*n+1, j)*binomial(10-j, 2)^n.
G.f. and e.g.f. are in the file "Generating functions". (End)

Terms a(11) and beyond from Andrew Howroyd, May 06 2020

A151631 Number of permutations of 2 indistinguishable copies of 1..n with exactly 9 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 0, 0, 0, 0, 716, 6030140, 6727188848, 2898916824320, 713977455470200, 121976270685699352, 16151017610840330800, 1776999565259831468176, 170177528415687895764196, 14652450038892876986833140, 1161079239300986084649361440, 86154881092488160155801056400

Offset: 1

R. H. Hardin, May 29 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
G. C. Greubel, Generating functions
Index entries for linear recurrences with constant coefficients, signature (715, -246246, 54441244, -8688122872, 1067021500896, -104995425911880, 8509695500071416, -579576260744202564, 33677542872235956572, -1689382985951061375584, 73851002736478171785480, -2834880964029477987484784, 96157307046055530022659296, -2897042624210196682636584168, 77864265942445209589091609880, -1873766333805076180258298532750, 40496615572103854298268667369290, -788076483208734925614168894880060, 13838951374046160498670120531900480, -219687004175759538500502785436424080, 3157288221085995379662504956708878000, -41129275275200013601737955696226964280, 486094673693240626024143992242097156520, -5215868577535025043910503669088440816660, 50836490264499322636673534370758109300396, -450178157479731675331980353762821217397840, 3622293207037319736887764557341829467018616, -26480103842294874673252984165092468826819824, 175815364714697377375288287523495537964680512, -1059685472384814743761882978983528331243780536, 5794006095745855069703806488744115029544310280, -28712845744794238874234856225294257655712864281, 128823074763417157330327450612996297705760524899, -522591702536253659265150517358362904804073702702, 1913877482958449061349927143878484707275902379044, -6316428749606413461495948122861485717837380579480, 18747314973123796053526567679937869016617533512144, -49921577593788411541183731318082844750585106215136, 118943703030207917584554870579506771381266070945088, -252783808588069216519093758900774170980091669600640, 477488323849250418683533669723998946360819532025600, -798356115352276242760013175029559061134250770240000, 1175928120255423597428632028457343135596494440320000, -1517382017920428119910882859627104065334963129600000, 1704039910436999395713170098091040014017105344000000, -1652387285823260958429823161445909045075804800000000, 1370326851226558633202522928960523884556128000000000, -960372144298001827617428397061627444681920000000000, 560220235705759309094565929790235607973120000000000, -266618977212745148402028435378503712537600000000000, 100714095994301538183255879386079220224000000000000, -29009478229516455744577541121897523200000000000000, 5977866876033168754616374301599948800000000000000, -783993082018895079509762803666944000000000000000, 49121618545275670528799969525760000000000000000).

Column k=9 of A154283.

[(&+[(-1)^j*Binomial(2*n+1, j)*Binomial(11-j, 2)^n: j in [0..9]]): n in [1..30]]; // G. C. Greubel, Sep 08 2022

With[{B=Binomial}, Table[Sum[(-1)^j*B[2n+1, j]*B[11-j, 2]^n, {j,0,9}], {n, 30}]] (* G. C. Greubel, Sep 08 2022 *)

def A151631(n): return sum((-1)^j*binomial(2*n+1, j)*binomial(11-j, 2)^n for j in (0..9))
[A151631(n) for n in (1..30)] # G. C. Greubel, Sep 08 2022

From G. C. Greubel, Sep 08 2022: (Start)
a(n) = Sum_{j=0..9} (-1)^j*binomial(2*n+1, j)*binomial(11-j, 2)^n.
G.f. and e.g.f. are in the file "Generating functions". (End)

Terms a(11) and beyond from Andrew Howroyd, May 06 2020

cp's OEIS Frontend

A174264 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^(3*n+1)*Sum_{k >= 0} (k*(k+1)*(2*k+1)/6)^n*x^k and p(0, x) = 1, read by rows.

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A151626 Number of permutations of 2 indistinguishable copies of 1..n with exactly 4 adjacent element pairs in decreasing order.

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A151627 Number of permutations of 2 indistinguishable copies of 1..n with exactly 5 adjacent element pairs in decreasing order.

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A151628 Number of permutations of 2 indistinguishable copies of 1..n with exactly 6 adjacent element pairs in decreasing order.

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A151629 Number of permutations of 2 indistinguishable copies of 1..n with exactly 7 adjacent element pairs in decreasing order.

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A151630 Number of permutations of 2 indistinguishable copies of 1..n with exactly 8 adjacent element pairs in decreasing order.

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A151631 Number of permutations of 2 indistinguishable copies of 1..n with exactly 9 adjacent element pairs in decreasing order.

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A174264 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)(1-x)^(3n+1)Sum_{k >= 0} (k(k+1)(2k+1)/6)^n*x^k and p(0, x) = 1, read by rows.