{"id":"n-8a857e84342335b8","revision":1,"first_index":0,"fingerprint":"f6-08a7d3764c1b9874","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["289","510","225"],["4913","13005","11475","3375"],["83521","294780","390150","229500","50625"],["1419857","6264075","11054250","9753750","4303125","759375"],["24137569","127787130","281883375","331627500","219459375","77456250","11390625"],["410338673","2534444745","6708824325","9865918125","8705221875","4608646875","1355484375","170859375"],["6975757441","49240640760","152066684700","268352973000","295977543750","208925325000","92172937500","23236875000","2562890625"],["118587876497","941727254535","3323743251300","6843000811500","9056912838750","7991393681250","4700819812500","1777620937500","392122265625","38443359375"],["2015993900449","17788181474550","70629544090125","166187162565000","256612530431250","271707385162500","199784842031250","100731853125000","33330392578125","6535371093750","576650390625"],["34271896307633","332638993574085","1467524971650375","3884624924956875","6855220455806250","8468213504231250","7471953091968750","4709214133593750","2077594470703125","611057197265625","107833623046875","8649755859375"],["582622237229761","6168941335373940","29937509421667650","88051498299022500","174808121623059375","246787936409025000","254046405126937500","192135936650625000","105957318005859375","41551889414062500","10999029550781250","1764550195312500","129746337890625"],["9904578032905937","113611336259803395","601471780198959150","1945938112408397250","4292510542077346875","6817516743299315625","8020607933293312500","7077006999964687500","4683313455858984375","2295741890126953125","810261843574218750","194982796582031250","28673940673828125","1946195068359375"],["168377826559400929","2079961386910246770","11929190307279356475","42103024613927140500","102161750901440855625","180285442767248568750","238613086015476046875","240618237998799375000","185771433749073046875","109277313970042968750","48210579692666015625","15468635195507812500","3412198940185546875","463194426269531250","29192926025390625"],["2862423051509815793","37885010975865209025","233995656027402761625","894689273045951735625","2368295134533401653125","4597278790564838503125","6760704103771821328125","7669706336211730078125","6767387943716232421875","4644285843726826171875","2458739564325966796875","986125493713623046875","290036909915771484375","59057289349365234375","7444196136474609375","437893890380859375"],["48661191875666868481","686981532362355790320","4546201317103825083000","18719652482192220930000","53681356382757104137500","113678166457603279350000","183891151622593540125000","231795569272176731250000","230091190086351902343750","180463678499099531250000","111462860249443828125000","53645226858021093750000","19722509874272460937500","5354527567675781250000","1012410674560546875000","119107138183593750000","6568408355712890625"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1024","32768","1048576","33554432","1073741824","34359738368","1099511627776","35184372088832","1125899906842624","36028797018963968","1152921504606846976","36893488147419103232","1180591620717411303424","37778931862957161709568","1208925819614629174706176"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["289","510","225"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["17","15"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167576,"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-633275fbd6b7b077"]},{"id":167577,"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-633275fbd6b7b077"]},{"id":167578,"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-633275fbd6b7b077"]}],"region_exact":{},"incoming_edges":[{"id":"e-633275fbd6b7b077","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-8a857e84342335b8","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":"298464d3d1e0cde2bb51f9a8c516cac179bd19b2dbf86b0e56d9dbc71b5b8c77","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-a5302200d285579d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.","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":[{"id":"l-0fe78822a191002e","kind":"oeis_match","node_a_id":"n-8a857e84342335b8","node_b_id":null,"fingerprint":null,"oeis_id":"A009976","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A009976","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009976","relation":"match_row_sums"}],"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:            289,             510,              225;\n  1:           4913,           13005,            11475,             3375;\n  2:          83521,          294780,           390150,           229500,            50625;\n  3:        1419857,         6264075,         11054250,          9753750,          4303125,           759375;\n  4:       24137569,       127787130,        281883375,        331627500,        219459375,         77456250,         11390625;\n  5:      410338673,      2534444745,       6708824325,       9865918125,       8705221875,       4608646875,       1355484375,        170859375;\n  6:     6975757441,     49240640760,     152066684700,     268352973000,     295977543750,     208925325000,      92172937500,      23236875000,       2562890625;\n  7:   118587876497,    941727254535,    3323743251300,    6843000811500,    9056912838750,    7991393681250,    4700819812500,    1777620937500,     392122265625,     38443359375;\n  8:  2015993900449,  17788181474550,   70629544090125,  166187162565000,  256612530431250,  271707385162500,  199784842031250,  100731853125000,   33330392578125,   6535371093750,    576650390625;\n  9: 34271896307633, 332638993574085, 1467524971650375, 3884624924956875, 6855220455806250, 8468213504231250, 7471953091968750, 4709214133593750, 2077594470703125, 611057197265625, 107833623046875, 8649755859375;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (17+15*x)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 510*x + 225*x^2.","mathematica":"P[0, x_] := 289 + 510*x + 225*x^2;\nP[n_, x_] := P[n, x] = Expand[(17 + 15*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 289 + 510*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 15*x)*v[i - 1]); 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-a5302200d285579d","steps":["basis A013610"],"edges":["e-633275fbd6b7b077"],"text":"A038331 \u2192 basis A013610"},"headline":"A038331 \u2192 basis A013610","chain":null}
