{"id":"n-1912def36a8ae2c1","revision":0,"first_index":0,"fingerprint":"f6-c4264000c5d1dd22","name":"basis_map_A013609 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","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":true,"rows":[["1"],["22","20"],["484","880","400"],["10648","29040","26400","8000"],["234256","851840","1161600","704000","160000"],["5153632","23425600","42592000","38720000","17600000","3200000"],["113379904","618435840","1405536000","1703680000","1161600000","422400000","64000000"],["2494357888","15873186560","43290508800","65591680000","59628800000","32524800000","9856000000","1280000000"],["54875873536","399097262080","1269854924800","2308827136000","2623667200000","1908121600000","867328000000","225280000000","25600000000"],["1207269217792","9877657236480","35918753587200","76191295488000","103897221120000","94452019200000","57243648000000","22302720000000","5068800000000","512000000000"],["26559922791424","241453843558400","987765723648000","2394583572480000","3809564774400000","4155888844800000","3148400640000000","1635532800000000","557568000000000","112640000000000","10240000000000"],["584318301411328","5843183014113280","26559922791424000","72436153067520000","131702096486400000","167620850073600000","152382590976000000","98949734400000000","44977152000000000","13629440000000000","2478080000000000","204800000000000"],["12855002631049216","140236392338718720","701181961693593600","2124793823313920000","4346169184051200000","6321700631347200000","6704834002944000000","5224545976320000000","2968492032000000000","1199390720000000000","327106560000000000","54067200000000000","4096000000000000"],["282810057883082752","3342300684072796160","18230731004033433600","60769103346778112000","138111598515404800000","226000797570662400000","273940360691712000000","249036691537920000000","169797744230400000000","85756436480000000000","31184158720000000000","7731609600000000000","1171456000000000000","81920000000000000"],["6221821273427820544","79186816207263170560","467922095770191462400","1701534893709787136000","4253837234274467840000","7734249516862668800000","10546703886630912000000","10957614427668480000000","8716284203827200000000","5282596487168000000000","2401180221440000000000","793778585600000000000","180404224000000000000","25231360000000000000","1638400000000000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","42","1764","74088","3111696","130691232","5489031744","230539333248","9682651996416","406671383849472","17080198121677824","717368321110468608","30129469486639681536","1265437718438866624512","53148384174432398229504"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["22","20"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200868,"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-8ba0ed0f0294a78a"]},{"id":200869,"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-8ba0ed0f0294a78a"]},{"id":200870,"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-8ba0ed0f0294a78a"]}],"region_exact":{},"incoming_edges":[{"id":"e-8ba0ed0f0294a78a","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-1912def36a8ae2c1","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":"7011445dadca7c1619d91f370a3005fdd4dd529eee68beaec0e00135c1c449bc","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-4f44de8e265280a7","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(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-0550f11958f6eb2b","kind":"fingerprint_related","node_a_id":"n-7d2bbb701a5b14c4","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-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-0bd87802646d32a7","kind":"fingerprint_related","node_a_id":"n-4d5b6d358f445bab","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-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-1cb540fc2a33cdcc","kind":"fingerprint_related","node_a_id":"n-e1d1088f7d1ff894","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-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-620e411a7157c3f9","kind":"fingerprint_related","node_a_id":"n-84ca83da96405582","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-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-724f3ac82c02d125","kind":"fingerprint_related","node_a_id":"n-f421d7ccf2f25523","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-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-7f85bf1f78e01c36","kind":"fingerprint_related","node_a_id":"n-c4b5ec42f7a2f0ec","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-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-a335cf9863d72239","kind":"fingerprint_related","node_a_id":"n-1912def36a8ae2c1","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-c1c55622d621a692","kind":"fingerprint_related","node_a_id":"n-20b892927bf76ac2","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-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-e56c2baca0c1a3a6","kind":"fingerprint_related","node_a_id":"n-5bdb8d0739751ee3","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-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-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-013ff8ec0291cf3e","other":{"id":"n-013ff8ec0291cf3e","name":"basis_map_A013609 of basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.","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":[["121","220","100"],["1331","3630","3300","1000"],["14641","53240","72600","44000","10000"]],"card_rows":[["121","220","100"],["1331","3630","3300","1000"],["14641","53240","72600","44000","10000"],["161051","732050","1331000","1210000","550000","100000"]],"row_sums_preview":["441","9261","194481","4084101","85766121","1801088541","37822859361","794280046581"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}}],"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:            22,            20;\n  2:           484,           880,            400;\n  3:         10648,         29040,          26400,           8000;\n  4:        234256,        851840,        1161600,         704000,          160000;\n  5:       5153632,      23425600,       42592000,       38720000,        17600000,        3200000;\n  6:     113379904,     618435840,     1405536000,     1703680000,      1161600000,      422400000,       64000000;\n  7:    2494357888,   15873186560,    43290508800,    65591680000,     59628800000,    32524800000,     9856000000,     1280000000;\n  8:   54875873536,  399097262080,  1269854924800,  2308827136000,   2623667200000,  1908121600000,   867328000000,   225280000000,   25600000000;\n  9: 1207269217792, 9877657236480, 35918753587200, 76191295488000, 103897221120000, 94452019200000, 57243648000000, 22302720000000, 5068800000000, 512000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (22+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(22 + 20*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (22 + 20*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-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}