{"id":"n-013ff8ec0291cf3e","revision":0,"first_index":0,"fingerprint":"f6-c4264000c5d1dd22","name":"basis_map_A013609 of basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.","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":[["121","220","100"],["1331","3630","3300","1000"],["14641","53240","72600","44000","10000"],["161051","732050","1331000","1210000","550000","100000"],["1771561","9663060","21961500","26620000","18150000","6600000","1000000"],["19487171","124009270","338207100","512435000","465850000","254100000","77000000","10000000"],["214358881","1558973680","4960370800","9018856000","10248700000","7453600000","3388000000","880000000","100000000"],["2357947691","19292299290","70153815600","148811124000","202924260000","184476600000","111804000000","43560000000","9900000000","1000000000"],["25937424601","235794769100","964614964500","2338460520000","3720278100000","4058485200000","3074610000000","1597200000000","544500000000","110000000000","10000000000"],["285311670611","2853116706110","12968712300500","35369215365000","64307664300000","81846118200000","74405562000000","48315300000000","21961500000000","6655000000000","1210000000000","100000000000"],["3138428376721","34237400473320","171187002366600","518748492020000","1061076460950000","1543383943200000","1636922364000000","1275523920000000","724729500000000","292820000000000","79860000000000","13200000000000","1000000000000"],["34522712143931","407995688973730","2225431030765800","7418103435886000","16859325990650000","27587987984700000","33439985436000000","30399986760000000","20727263700000000","10468315000000000","3806660000000000","943800000000000","143000000000000","10000000000000"],["379749833583241","4833179700150340","28559698228161100","103853448102404000","259633620256010000","472061127738200000","643719719643000000","668799708720000000","531999768300000000","322424102000000000","146556410000000000","48448400000000000","11011000000000000","1540000000000000","100000000000000"],["4177248169415651","56962475037486150","362488477511275500","1427984911408055000","3894504303840150000","7789008607680300000","11801528193455000000","13793993992350000000","12539994538500000000","8866662805000000000","4836361530000000000","1998496500000000000","605605000000000000","127050000000000000","16500000000000000","1000000000000000"],["45949729863572161","668359707106504160","4556998002998892000","19332718800601360000","57119396456322200000","124624137722884800000","207706896204808000000","269749215850400000000","275879879847000000000","222933236240000000000","141866604880000000000","70347076800000000000","26646620000000000000","7453600000000000000","1452000000000000000","176000000000000000","10000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["441","9261","194481","4084101","85766121","1801088541","37822859361","794280046581","16679880978201","350277500542221","7355827511386641","154472377739119461","3243919932521508681","68122318582951682301","1430568690241985328321"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["121","220","100"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["11","10"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200807,"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-dd9bb2c38d133d21"]},{"id":200808,"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-dd9bb2c38d133d21"]},{"id":200809,"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-dd9bb2c38d133d21"]}],"region_exact":{},"incoming_edges":[{"id":"e-dd9bb2c38d133d21","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-013ff8ec0291cf3e","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":"3c58762b5bbe6ff81bbe155f73bb18225aabdde2ee45c8ed9af766050331b179","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-4550174e70f7b7f7","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.","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-06c692b23f97bcad","kind":"fingerprint_related","node_a_id":"n-c4b5ec42f7a2f0ec","node_b_id":"n-013ff8ec0291cf3e","fingerprint":"f6-c4264000c5d1dd22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c4264000c5d1dd22"},"other_id":"n-c4b5ec42f7a2f0ec","other":{"id":"n-c4b5ec42f7a2f0ec","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.","first_index":0,"fingerprint":"f6-c4264000c5d1dd22","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["484","880","400"],["10648","29040","26400","8000"],["234256","851840","1161600","704000","160000"]],"card_rows":[["484","880","400"],["10648","29040","26400","8000"],["234256","851840","1161600","704000","160000"],["5153632","23425600","42592000","38720000","17600000","3200000"]],"row_sums_preview":["1764","74088","3111696","130691232","5489031744","230539333248","9682651996416","406671383849472"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["basis A013609"],"edges":["e-0c8cc5263f914382"],"text":"A038336 \u2192 basis A013609"},"headline":"A038336 \u2192 basis A013609","chain":null}},{"id":"l-246ebcda354ae1ec","kind":"fingerprint_related","node_a_id":"n-4d5b6d358f445bab","node_b_id":"n-013ff8ec0291cf3e","fingerprint":"f6-c4264000c5d1dd22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c4264000c5d1dd22"},"other_id":"n-4d5b6d358f445bab","other":{"id":"n-4d5b6d358f445bab","name":"derivative of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","first_index":0,"fingerprint":"f6-c4264000c5d1dd22","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["10"],["220","200"]],"card_rows":[["10"],["220","200"],["3630","6600","3000"],["53240","145200","132000","40000"]],"row_sums_preview":["0","10","420","13230","370440","9724050","245046060","6003628470"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e1d1088f7d1ff894","steps":["D"],"edges":["e-3496344452743c97"],"text":"A038324 \u2192 D"},"headline":"A038324 \u2192 D","chain":null}},{"id":"l-31d0eb49ce5037e4","kind":"fingerprint_related","node_a_id":"n-013ff8ec0291cf3e","node_b_id":"n-4cca32ac9f07ae5b","fingerprint":"f6-c4264000c5d1dd22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c4264000c5d1dd22"},"other_id":"n-4cca32ac9f07ae5b","other":{"id":"n-4cca32ac9f07ae5b","name":"basis_map_A013609 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","first_index":0,"fingerprint":"f6-c4264000c5d1dd22","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["40","44"],["1600","3520","1936"]],"card_rows":[["1"],["40","44"],["1600","3520","1936"],["64000","211200","232320","85184"]],"row_sums_preview":["1","84","7056","592704","49787136","4182119424","351298031616","29509034655744"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b7cf68162d40a956","steps":["basis A013609","basis A013609"],"edges":["e-59f75239fe41171d","e-21151881b7f80294"],"text":"A038277 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A038277 \u2192 basis A013609 \u2192 basis A013609","chain":null}},{"id":"l-36ed37e43185d3dd","kind":"fingerprint_related","node_a_id":"n-5bdb8d0739751ee3","node_b_id":"n-013ff8ec0291cf3e","fingerprint":"f6-c4264000c5d1dd22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c4264000c5d1dd22"},"other_id":"n-5bdb8d0739751ee3","other":{"id":"n-5bdb8d0739751ee3","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","first_index":0,"fingerprint":"f6-c4264000c5d1dd22","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["880","968"],["26400","58080","31944"],["704000","2323200","2555520","937024"]],"card_rows":[["880","968"],["26400","58080","31944"],["704000","2323200","2555520","937024"],["17600000","77440000","127776000","93702400","25768160"]],"row_sums_preview":["1848","116424","6519744","342286560","17251242624","845310888576","40574922651648","1917165095290368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cfc5b48d32898a3e","steps":["basis A013609","D"],"edges":["e-21c2803b3d0372c9","e-0fca4daed101d063"],"text":"A038301 \u2192 basis A013609 \u2192 D"},"headline":"A038301 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-591e23fea2469a1e","kind":"fingerprint_related","node_a_id":"n-84ca83da96405582","node_b_id":"n-013ff8ec0291cf3e","fingerprint":"f6-c4264000c5d1dd22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c4264000c5d1dd22"},"other_id":"n-84ca83da96405582","other":{"id":"n-84ca83da96405582","name":"reversal of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","first_index":0,"fingerprint":"f6-c4264000c5d1dd22","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","11"],["100","220","121"]],"card_rows":[["1"],["10","11"],["100","220","121"],["1000","3300","3630","1331"]],"row_sums_preview":["1","21","441","9261","194481","4084101","85766121","1801088541"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e1d1088f7d1ff894","steps":["reverse"],"edges":["e-13ae4798662a2199"],"text":"A038324 \u2192 reverse"},"headline":"A038324 \u2192 reverse","chain":null}},{"id":"l-74fd912b3e56d2f9","kind":"fingerprint_related","node_a_id":"n-20b892927bf76ac2","node_b_id":"n-013ff8ec0291cf3e","fingerprint":"f6-c4264000c5d1dd22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c4264000c5d1dd22"},"other_id":"n-20b892927bf76ac2","other":{"id":"n-20b892927bf76ac2","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.","first_index":0,"fingerprint":"f6-c4264000c5d1dd22","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["880","800"],["29040","52800","24000"],["851840","2323200","2112000","640000"]],"card_rows":[["880","800"],["29040","52800","24000"],["851840","2323200","2112000","640000"],["23425600","85184000","116160000","70400000","16000000"]],"row_sums_preview":["1680","105840","5927040","311169600","15682947840","768464444160","36886293319680","1742877359354880"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["basis A013609","D"],"edges":["e-0c8cc5263f914382","e-dc15e5f1765bfe25"],"text":"A038336 \u2192 basis A013609 \u2192 D"},"headline":"A038336 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-854d9763013dcd80","kind":"fingerprint_related","node_a_id":"n-7d2bbb701a5b14c4","node_b_id":"n-013ff8ec0291cf3e","fingerprint":"f6-c4264000c5d1dd22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c4264000c5d1dd22"},"other_id":"n-7d2bbb701a5b14c4","other":{"id":"n-7d2bbb701a5b14c4","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","first_index":0,"fingerprint":"f6-c4264000c5d1dd22","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","880","484"],["8000","26400","29040","10648"],["160000","704000","1161600","851840","234256"]],"card_rows":[["400","880","484"],["8000","26400","29040","10648"],["160000","704000","1161600","851840","234256"],["3200000","17600000","38720000","42592000","23425600","5153632"]],"row_sums_preview":["1764","74088","3111696","130691232","5489031744","230539333248","9682651996416","406671383849472"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cfc5b48d32898a3e","steps":["basis A013609"],"edges":["e-21c2803b3d0372c9"],"text":"A038301 \u2192 basis A013609"},"headline":"A038301 \u2192 basis A013609","chain":null}},{"id":"l-a204757c8617395a","kind":"fingerprint_related","node_a_id":"n-f421d7ccf2f25523","node_b_id":"n-013ff8ec0291cf3e","fingerprint":"f6-c4264000c5d1dd22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c4264000c5d1dd22"},"other_id":"n-f421d7ccf2f25523","other":{"id":"n-f421d7ccf2f25523","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","first_index":0,"fingerprint":"f6-c4264000c5d1dd22","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","11"],["100","220","121"]],"card_rows":[["1"],["10","11"],["100","220","121"],["1000","3300","3630","1331"]],"row_sums_preview":["1","21","441","9261","194481","4084101","85766121","1801088541"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["x \u21a6 x/(x+1)"],"edges":["e-922005d5e5465f35"],"text":"A038303 \u2192 x \u21a6 x/(x+1)"},"headline":"A038303 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-af1c3872e3326d1b","kind":"fingerprint_related","node_a_id":"n-e1d1088f7d1ff894","node_b_id":"n-013ff8ec0291cf3e","fingerprint":"f6-c4264000c5d1dd22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c4264000c5d1dd22"},"other_id":"n-e1d1088f7d1ff894","other":{"id":"n-e1d1088f7d1ff894","name":"Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","first_index":0,"fingerprint":"f6-c4264000c5d1dd22","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","10"],["121","220","100"]],"card_rows":[["1"],["11","10"],["121","220","100"],["1331","3630","3300","1000"]],"row_sums_preview":["1","21","441","9261","194481","4084101","85766121","1801088541"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A038324"],"derivation":{"root":"n-e1d1088f7d1ff894","steps":[],"edges":[],"text":"A038324"},"headline":"A038324: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","chain":null}},{"id":"l-ef42a4007bc54c7d","kind":"fingerprint_related","node_a_id":"n-013ff8ec0291cf3e","node_b_id":"n-1912def36a8ae2c1","fingerprint":"f6-c4264000c5d1dd22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c4264000c5d1dd22"},"other_id":"n-1912def36a8ae2c1","other":{"id":"n-1912def36a8ae2c1","name":"basis_map_A013609 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-c4264000c5d1dd22","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","20"],["484","880","400"]],"card_rows":[["1"],["22","20"],["484","880","400"],["10648","29040","26400","8000"]],"row_sums_preview":["1","42","1764","74088","3111696","130691232","5489031744","230539333248"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013609","basis A013609"],"edges":["e-e169cb33ba7b2c8c","e-8ba0ed0f0294a78a"],"text":"A038271 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A038271 \u2192 basis A013609 \u2192 basis A013609","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:          121,           220,            100;\n  1:         1331,          3630,           3300,           1000;\n  2:        14641,         53240,          72600,          44000,          10000;\n  3:       161051,        732050,        1331000,        1210000,         550000,         100000;\n  4:      1771561,       9663060,       21961500,       26620000,       18150000,        6600000,        1000000;\n  5:     19487171,     124009270,      338207100,      512435000,      465850000,      254100000,       77000000,       10000000;\n  6:    214358881,    1558973680,     4960370800,     9018856000,    10248700000,     7453600000,     3388000000,      880000000,      100000000;\n  7:   2357947691,   19292299290,    70153815600,   148811124000,   202924260000,   184476600000,   111804000000,    43560000000,     9900000000,    1000000000;\n  8:  25937424601,  235794769100,   964614964500,  2338460520000,  3720278100000,  4058485200000,  3074610000000,  1597200000000,   544500000000,  110000000000,   10000000000;\n  9: 285311670611, 2853116706110, 12968712300500, 35369215365000, 64307664300000, 81846118200000, 74405562000000, 48315300000000, 21961500000000, 6655000000000, 1210000000000, 100000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (11+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 220*x + 100*x^2.","mathematica":"P[0, x_] := 121 + 220*x + 100*x^2;\nP[n_, x_] := P[n, x] = Expand[(11 + 10*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] = 121 + 220*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 10*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-bbfc5a54ff13a785","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-7ae18e4d322020fd","e-dd9bb2c38d133d21"],"text":"A013612 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A013612 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}
