{"id":"n-2f014dad76c8185f","revision":0,"first_index":0,"fingerprint":"f6-b0048c17ed6c5e8a","name":"basis_map_A013610 of Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +3)*prime(j)*T(n-2, k-1) with j=9, read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2"],["46","69"],["1058","3174","18"],["24334","109503","109539","54"],["559682","3358092","5155407","354807","162"],["12872686","96545145","231962274","174923928","1142991","486"],["296071778","2664646002","8109496872","8687291202","1041251355","3664683","1458"],["6809650894","71501334387","276144625197","462304665270","282161212641","4787512128","11701179","4374"],["156621970562","1879463646744","8544752299629","17699665824225","15042045777255","2366601917463","19696768623","37224927","13122"],["3602305322926","48631121859501","259711183036404","684460303783200","889346414709018","461023306641324","14081956032564","76121876736","118038951","39366"],["82853022427298","1242795336409470","7515773916504174","23078338831938708","36744197630483826","26529078914786826","4849953549637896","70956709225632","282546965259","373209363","118098"],["1905619515827854","31442722011159591","214666930823236839","775619830360424934","1564060710243096270","1674627962742151704","764111499930643158","35133295225013676","323792833811211","1019444768280","1176905619","354294"],["43829248864040642","788926479552731556","5954591271808790523","24232434514031504475","56642941334570343192","74097325538626693050","47335269800088956070","9424455874223248386","209000213194824315","1383008203752813","3601525524399","3702549447","1062882"],["1008072723872934766","19657418115522227937","163520593764069416310","752005403670444803352","2065297643450906427585","3393292750227670183938","3112564288695936455700","1285021666141896906432","79755904768835025225","1099434864007502550","5633971576958934","12517489038024","11623146111","3188646"],["23185672649077499618","486899125630627491978","4405235495030259363684","22327041211761251984886","68857949773342273011237","130847108659757179678989","146634966846987927474474","85048554325950329495832","17771560472685947825205","544975885916233121889","5315290959242536272","22147348525189902","42941220927003","36415931643","9565938"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","115","4250","243430","9428150","517447510","20802423350","1103720706070","45708885442550","2360932831339990","100114230831706550","5061893384307393430","218730160888394021750","10873228822923854215510","476952814950996521098550"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["46","69"],["1058","3174","18"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-966","-2898"],["483","1449"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","-6"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203777,"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-d081617b3e93bc33"]},{"id":203778,"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-d081617b3e93bc33"]},{"id":203779,"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-d081617b3e93bc33"]},{"id":203780,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-d081617b3e93bc33"]}],"region_exact":{},"incoming_edges":[{"id":"e-d081617b3e93bc33","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-2f014dad76c8185f","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":"b3371f71509f4b611e9fca52718194395d0d6762cf8268797780785a3ab8514c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6e554ebca19f83a9","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +3)*prime(j)*T(n-2, k-1) with j=9, read by rows.","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:             2;\n  1:            46,             69;\n  2:          1058,           3174,              18;\n  3:         24334,         109503,          109539,              54;\n  4:        559682,        3358092,         5155407,          354807,             162;\n  5:      12872686,       96545145,       231962274,       174923928,         1142991,             486;\n  6:     296071778,     2664646002,      8109496872,      8687291202,      1041251355,         3664683,           1458;\n  7:    6809650894,    71501334387,    276144625197,    462304665270,    282161212641,      4787512128,       11701179,        4374;\n  8:  156621970562,  1879463646744,   8544752299629,  17699665824225,  15042045777255,   2366601917463,    19696768623,    37224927,     13122;\n  9: 3602305322926, 48631121859501, 259711183036404, 684460303783200, 889346414709018, 461023306641324, 14081956032564, 76121876736, 118038951, 39366;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (483*n-966+(1449*n-2898)*x)*P(n-2,x) + (2*n-4+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 46 + 69*x, P(2,x) = 1058 + 3174*x + 18*x^2.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 46 + 69*x;\nP[2, x_] := 1058 + 3174*x + 18*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((483*n - 966 + (1449*n - 2898)*x)*P[n - 2, x] + (2*n - 4 + (3*n - 6)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 46 + 69*x; v[3] = 1058 + 3174*x + 18*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((483*n - 966 + (1449*n - 2898)*x)*v[i - 2] + (2*n - 4 + (3*n - 6)*x)*v[i - 1])/(n - 2)); 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-6e554ebca19f83a9","steps":["basis A013610"],"edges":["e-d081617b3e93bc33"],"text":"A153656 \u2192 basis A013610"},"headline":"A153656 \u2192 basis A013610","chain":null}
