{"id":"n-207567e2aa72e118","revision":0,"first_index":0,"fingerprint":"f6-25b337cc6c0a5b46","name":"basis_map_A013610 of Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] ordered by the product order, 0 <= k <= 2n, n>=0.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","15","18"],["26","225","711","972","486"],["252","3096","15579","41121","60102","46170","14580"],["2568","40860","280962","1090989","2618001","3977910","3739770","1990170","459270"],["26928","526032","4580712","23423472","77917707","176249115","274655124","291269034","201252114","81841914","14880348"],["287648","6661200","70164288","444621960","1888318170","5663924433","12306201615","19520536518","22441769280","18240826590","9952472754","3273676560","491051484"],["3112896","83362560","1029909456","7781957712","40183597476","150033369096","417787476075","881610018381","1416940407612","1726430020956","1570042169298","1033604915310","465727257468","128585338596","16415149608"],["34013312","1034064576","14656159584","128567836656","781409312712","3489593151492","11846456011026","31189980384549","64373878213281","104515498710252","133059803969736","131455873579338","98812080393642","54637825470228","20962146049416","4986101693430","554011299270"],["374416128","12738988800","203701485312","2034299556864","14220017319456","73891598887008","295840988467920","933250753183392","2351768529036339","4770160158570471","7807171270773294","10287165484508220","10833078522490686","8994580456924062","5759326372281768","2744489900835540","916642473047730","191503239114330","18836384175180"],["4145895936","156073699584","2779082037504","31124870970624","245923019504064","1457272807803360","6720522393342528","24701715349265808","73500074045224218","178808433353950113","357632415412930791","589162435083046440","798107024112489630","884270450610557010","793565436252418830","568012051220631732","316698195446151948","132572355463334358","39200399106967098","7300982506299768","644204338791156"],["46127840256","1903557888000","37349959216896","463576555434240","4084691981132160","27180770328861696","141856508928278880","595230403470718176","2042012431172943180","5793057496450035000","13691169654794852283","27069298930837727865","44836067592952254738","62136461576122552710","71759297184075924930","68561426833643313882","53589152568854593080","33701083184011493952","16637213205347156370","6209104955407757910","1646996238165789108","276715045526201100","22137203642096088"],["515268544512","23129686195200","495793164667392","6754156955783424","65654468986563072","484675835156295552","2823596950521948480","13316089497350425632","51738552806124719304","167678689595199416172","457183856903440619106","1054657611530969384925","2065141765740775894641","3436229749515394659282","4853847730174088969466","5802368882751392804424","5837612163317445319380","4901239195064047104984","3392407522824185413548","1902188211758869888854","842434710595768381806","283620623217888296340","68208413955238396476","10437691517248305492","763733525652315036"],["5775088103424","280150440579072","6513233739429888","96614435428134912","1026950433784521984","8328184864033630464","53567275153032698112","280459386892241238528","1217176789000170719952","4436183269446902964144","13706101966943559556152","36136851687169553584656","81666960762169492676379","158605183573583329613331","264936706197939119894832","380353365607945398335478","468147833664400630686924","491866087552200886876956","438269077587331717014024","328120485100157401418448","203755955159574401316282","103068075286252917364986","41385301838824645236024","12690098347684295828556","2791211041328241513492","392147791056092528100","26436929734118597400"],["64912164888576","3383980053811200","84810435456098304","1360673508218738688","15699007006109885952","138726141613820557056","976354232781049424640","5618924191526859442944","26938480882827579313248","109053978748230606762672","376519107504370498956288","1116838112383605379524120","2861100259682579322668442","6352676814861108557651889","12250639636515509555822031","20532679610784043015899522","29891637214493161603775760","37723532461552171427656788","41127393588815668388364396","38535673555872691852065984","30808149417780551481029508","20808891887320864100918430","11715932519775770560169922","5397712841328924327144804","1982219304789154349732736","558013717658660563392300","113076413529196837922100","14683826115179015241600","917739132198688452600"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","36","2420","180900","14200500","1146622500","94300512500","7856258062500","660808950312500","55994177264062500","4772633954695312500","408740049351914062500","35144456395140820312500","3031914725363680664062500","262311031642032231445312500"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","15","18"],["26","225","711","972","486"]],"recurrence":{"denominator":{"coeffs":[["0"],["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-8","-24","-18"],["12","36","27"],["-4","-12","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["12","42","36"],["-30","-105","-90"],["12","42","36"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203102,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-253053961cc5c19e"]},{"id":203103,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-253053961cc5c19e"]},{"id":203104,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-253053961cc5c19e"]}],"region_exact":{},"incoming_edges":[{"id":"e-253053961cc5c19e","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-207567e2aa72e118","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"eff3877b2ecbfba32cc9eafac33dfd01dc6bdb60d124fe0b38cc43face11aad5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-97a17a1dc7998ff8","role":"sequence","name":"Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] ordered by the product order, 0 <= k <= 2n, n>=0.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:         1;\n  1:         3,          15,           18;\n  2:        26,         225,          711,           972,            486;\n  3:       252,        3096,        15579,         41121,          60102,          46170,           14580;\n  4:      2568,       40860,       280962,       1090989,        2618001,        3977910,         3739770,         1990170,           459270;\n  5:     26928,      526032,      4580712,      23423472,       77917707,      176249115,       274655124,       291269034,        201252114,         81841914,         14880348;\n  6:    287648,     6661200,     70164288,     444621960,     1888318170,     5663924433,     12306201615,     19520536518,      22441769280,      18240826590,       9952472754,        3273676560,         491051484;\n  7:   3112896,    83362560,   1029909456,    7781957712,    40183597476,   150033369096,    417787476075,    881610018381,    1416940407612,    1726430020956,    1570042169298,     1033604915310,      465727257468,     128585338596,      16415149608;\n  8:  34013312,  1034064576,  14656159584,  128567836656,   781409312712,  3489593151492,  11846456011026,  31189980384549,   64373878213281,  104515498710252,  133059803969736,   131455873579338,    98812080393642,   54637825470228,   20962146049416,    4986101693430,    554011299270;\n  9: 374416128, 12738988800, 203701485312, 2034299556864, 14220017319456, 73891598887008, 295840988467920, 933250753183392, 2351768529036339, 4770160158570471, 7807171270773294, 10287165484508220, 10833078522490686, 8994580456924062, 5759326372281768, 2744489900835540, 916642473047730, 191503239114330, 18836384175180;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(12*n^2-36*n+24)*x+(9*n^2-27*n+18)*x^2)*P(n-2,x) + (12*n^2-30*n+12+(42*n^2-105*n+42)*x+(36*n^2-90*n+36)*x^2)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 15*x + 18*x^2, P(2,x) = 26 + 225*x + 711*x^2 + 972*x^3 + 486*x^4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 15*x + 18*x^2;\nP[2, x_] := 26 + 225*x + 711*x^2 + 972*x^3 + 486*x^4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (12*n^2 - 36*n + 24)*x + (9*n^2 - 27*n + 18)*x^2)*P[n - 2, x] + (12*n^2 - 30*n + 12 + (42*n^2 - 105*n + 42)*x + (36*n^2 - 90*n + 36)*x^2)*P[n - 1, x])/(n^2 - 2*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 15*x + 18*x^2; v[3] = 26 + 225*x + 711*x^2 + 972*x^3 + 486*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (12*n^2 - 36*n + 24)*x + (9*n^2 - 27*n + 18)*x^2)*v[i - 2] + (12*n^2 - 30*n + 12 + (42*n^2 - 105*n + 42)*x + (36*n^2 - 90*n + 36)*x^2)*v[i - 1])/(n^2 - 2*n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-97a17a1dc7998ff8","steps":["basis A013610"],"edges":["e-253053961cc5c19e"],"text":"A316649 \u2192 basis A013610"},"headline":"A316649 \u2192 basis A013610","chain":null}