{"id":"n-2f533abae2b58c81","revision":0,"first_index":0,"fingerprint":"f6-74360a37e7e7e405","name":"basis_map_A013610 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(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":[["676","1560","900"],["17576","60840","70200","27000"],["456976","2109120","3650400","2808000","810000"],["11881376","68546400","158184000","182520000","105300000","24300000"],["308915776","2138647680","6169176000","9491040000","8213400000","3790800000","729000000"],["8031810176","64872312960","224558006400","431842320000","498279600000","344962800000","132678000000","21870000000"],["208827064576","1927634442240","7784677555200","17964640512000","25910539200000","23917420800000","13798512000000","4548960000000","656100000000"],["5429503678976","56383307435520","260230649702400","700620979968000","1212613234560000","1399169116800000","1076283936000000","532228320000000","153527400000000","19683000000000"],["141167095653376","1628851103692800","8457496115328000","26023064970240000","52546573497600000","72756794073600000","69958455840000000","46126454400000000","19958562000000000","5117580000000000","590490000000000"],["3670344486987776","46585141565614080","268760432109312000","930324572686080000","2146902860044800000","3468073850841600000","4001623674048000000","3298041489600000000","1902716244000000000","731813940000000000","168880140000000000","17714700000000000"],["95428956661682176","1321324015315599360","8385325481810534400","32251251853117440000","83729211541747200000","154577005923225600000","208084431050496000000","205797788951040000000","148411867032000000000","76108649760000000000","26345301840000000000","5526986400000000000","531441000000000000"],["2481152873203736576","37217293098056048640","257658182986541875200","1090092312635369472000","3144497055678950400000","6530878500256281600000","10047505385009664000000","11593275444241920000000","10032642211363200000000","6431180904720000000000","2968237340640000000000","934060701600000000000","179627058000000000000","15943230000000000000"],["64509974703297150976","1042084206745569361920","7815631550591770214400","36072145618115862528000","114459692826713794560000","264137752677031833600000","457161495017939712000000","602850323100579840000000","608646960822700800000000","468189969863616000000000","270109597998240000000000","113332698460800000000000","32692124556000000000000","5803335720000000000000","478296900000000000000"],["1677259342285725925376","29029488616483717939200","234468946517753106432000","1172344732588765532160000","4058116382038034534400000","10301372354404241510400000","19810331450777387520000000","29388953251153267200000000","33910330674407616000000000","30432348041135040000000000","21068548643862720000000000","11049938099928000000000000","4249976192280000000000000","1131650465400000000000000","186535791000000000000000","14348907000000000000000"],["43608742899428874059776","805084484297148444180480","6967077267956092305408000","37515031442840497029120000","140681367910651863859200000","389579172675651315302400000","824109788352339320832000000","1358422728053306572800000000","1763337195069196032000000000","1808550969301739520000000000","1460752705974481920000000000","919354849914009600000000000","441997523997120000000000000","156922197868800000000000000","38799444528000000000000000","5969145312000000000000000","430467210000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["3136","175616","9834496","550731776","30840979456","1727094849536","96717311574016","5416169448144896","303305489096114176","16985107389382393856","951166013805414055936","53265296773103187132416","2982856619293778479415296","167039970680451594847256576","9354238358105289311446368256"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["676","1560","900"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["26","30"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203974,"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-1cf4c56d3c91d5ed"]},{"id":203975,"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-1cf4c56d3c91d5ed"]},{"id":203976,"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-1cf4c56d3c91d5ed"]}],"region_exact":{},"incoming_edges":[{"id":"e-1cf4c56d3c91d5ed","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-2f533abae2b58c81","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":"97b27dad2118d81d9265b3a6baa3ebe7c694b9093753c21cafc16546098dd454","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-cf1de4f27da354f5","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(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-7dd48a70967f3821","kind":"fingerprint_related","node_a_id":"n-b17ced2e7aeac1db","node_b_id":"n-2f533abae2b58c81","fingerprint":"f6-74360a37e7e7e405","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-74360a37e7e7e405"},"other_id":"n-b17ced2e7aeac1db","other":{"id":"n-b17ced2e7aeac1db","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","first_index":0,"fingerprint":"f6-74360a37e7e7e405","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["13","15"],["169","390","225"]],"card_rows":[["1"],["13","15"],["169","390","225"],["2197","7605","8775","3375"]],"row_sums_preview":["1","28","784","21952","614656","17210368","481890304","13492928512"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-81e6fb4f9f226d77","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-ec56a67a9fb88114","e-7134e0a34ce6e3cb"],"text":"A038327 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038327 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-82953570f0323062","kind":"fingerprint_related","node_a_id":"n-d4932ac719fd68dc","node_b_id":"n-2f533abae2b58c81","fingerprint":"f6-74360a37e7e7e405","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-74360a37e7e7e405"},"other_id":"n-d4932ac719fd68dc","other":{"id":"n-d4932ac719fd68dc","name":"basis_map_A013609 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","first_index":0,"fingerprint":"f6-74360a37e7e7e405","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["26","30"],["676","1560","900"]],"card_rows":[["1"],["26","30"],["676","1560","900"],["17576","60840","70200","27000"]],"row_sums_preview":["1","56","3136","175616","9834496","550731776","30840979456","1727094849536"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e6ebe74518d42cbe","steps":["basis A013610","basis A013609"],"edges":["e-aef71e51dba5316b","e-ec5b4e8fb4d9f767"],"text":"A038259 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A038259 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-e2ef3814a00cfa7d","kind":"fingerprint_related","node_a_id":"n-e62a57d099bc4e7d","node_b_id":"n-2f533abae2b58c81","fingerprint":"f6-74360a37e7e7e405","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-74360a37e7e7e405"},"other_id":"n-e62a57d099bc4e7d","other":{"id":"n-e62a57d099bc4e7d","name":"basis_map_A013610 of basis_map_A007318 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-74360a37e7e7e405","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["13","15"],["169","390","225"]],"card_rows":[["1"],["13","15"],["169","390","225"],["2197","7605","8775","3375"]],"row_sums_preview":["1","28","784","21952","614656","17210368","481890304","13492928512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A007318","basis A013610"],"edges":["e-31008ec5d02c6da5","e-35212c222450aa91"],"text":"A013622 \u2192 basis A007318 \u2192 basis A013610"},"headline":"A013622 \u2192 basis A007318 \u2192 basis A013610","chain":null}}],"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:              676,              1560,                900;\n  1:            17576,             60840,              70200,              27000;\n  2:           456976,           2109120,            3650400,            2808000,              810000;\n  3:         11881376,          68546400,          158184000,          182520000,           105300000,            24300000;\n  4:        308915776,        2138647680,         6169176000,         9491040000,          8213400000,          3790800000,           729000000;\n  5:       8031810176,       64872312960,       224558006400,       431842320000,        498279600000,        344962800000,        132678000000,         21870000000;\n  6:     208827064576,     1927634442240,      7784677555200,     17964640512000,      25910539200000,      23917420800000,      13798512000000,       4548960000000,        656100000000;\n  7:    5429503678976,    56383307435520,    260230649702400,    700620979968000,    1212613234560000,    1399169116800000,    1076283936000000,     532228320000000,     153527400000000,     19683000000000;\n  8:  141167095653376,  1628851103692800,   8457496115328000,  26023064970240000,   52546573497600000,   72756794073600000,   69958455840000000,   46126454400000000,   19958562000000000,   5117580000000000,    590490000000000;\n  9: 3670344486987776, 46585141565614080, 268760432109312000, 930324572686080000, 2146902860044800000, 3468073850841600000, 4001623674048000000, 3298041489600000000, 1902716244000000000, 731813940000000000, 168880140000000000, 17714700000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (26+30*x)*P(n-1,x) for n >= 1, with P(0,x) = 676 + 1560*x + 900*x^2.","mathematica":"P[0, x_] := 676 + 1560*x + 900*x^2;\nP[n_, x_] := P[n, x] = Expand[(26 + 30*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] = 676 + 1560*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = (26 + 30*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-2bc1923905556bf6","steps":["basis A013609","basis A013610"],"edges":["e-772e88d8a50e6734","e-1cf4c56d3c91d5ed"],"text":"A038319 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A038319 \u2192 basis A013609 \u2192 basis A013610","chain":null}
