{"id":"n-bd9b770a0539d41d","revision":0,"first_index":0,"fingerprint":"f6-8b732aeffa021ba5","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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":[["144","408","289"],["1728","7344","10404","4913"],["20736","117504","249696","235824","83521"],["248832","1762560","4993920","7074720","5011260","1419857"],["2985984","25380864","89890560","169793280","180405360","102229704","24137569"],["35831808","355332096","1510161408","3565658880","5051350080","4293647568","2027555796","410338673"],["429981696","4873125888","24162582528","68460650496","121232401920","137396722176","97322678208","39392512608","6975757441"],["5159780352","65787199488","372794130432","1232291708928","2618619881472","3709711498752","3503616415488","2127195680832","753381803628","118587876497"],["61917364224","877162659840","5591911956480","21125000724480","52372397629440","89033075970048","105108492464640","85087827233280","45202908217680","14230545179640","2015993900449"],["743008370688","11578547109888","82014708695040","348562511953920","987593783869440","1958727671341056","2774864201066496","2807898298698240","1988927961577920","939215981856240","266111194859268","34271896307633"],["8916100448256","151573707620352","1181011805208576","5577000191262720","17776688109649920","40293826381873152","66596740825595904","80867471002509312","71601406616805120","45082367129099520","19160006029867296","4935153068299152","582622237229761"],["106993205379072","1970458199064576","16748894692048896","87001202983698432","308129260567265280","785729614446526464","1484155938398994432","2102554246065242112","2233963886444319744","1758212318034881280","996320313553099392","384941939327333856","90889069007842716","9904578032905937"],["1283918464548864","25464382880219136","234484525688684544","1328745645569212416","5176571577530056704","14666952803001827328","31167274706378883072","50461301905565810688","62550988820440952832","59075933885972011008","41845453169230174464","21556748602330695936","7634681796658788144","1663969109528197416","168377826559400929"],["15407021574586368","327399208459960320","3246708817227939840","19931184683538186240","84707534905037291520","264005150454032891904","623345494127577661440","1135379292875230740480","1608453998239910215680","1772278016579160330240","1506436314092286280704","970053687104881317120","458080907799527288640","149757219857537767440","30308008780692167220","2862423051509815793"],["184884258895036416","4190709868287492096","44526292350554603520","294368266095333212160","1355320558480596664320","4608089898834028658688","11968233487249491099648","24221424914671589130240","38602895957757845176320","48611054169028397629440","48205962050953160982528","37250061584827442577408","21987883574377309854720","9584462070882417116160","2909568842946448053120","549585225889884632256","48661191875666868481"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["841","24389","707281","20511149","594823321","17249876309","500246412961","14507145975869","420707233300201","12200509765705829","353814783205469041","10260628712958602189","297558232675799463481","8629188747598184440949","250246473680347348787521"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["144","408","289"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["12","17"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194310,"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-dd8524b7a1597679"]},{"id":194311,"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-dd8524b7a1597679"]},{"id":194312,"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-dd8524b7a1597679"]}],"region_exact":{},"incoming_edges":[{"id":"e-dd8524b7a1597679","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-bd9b770a0539d41d","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"b5b740dff66053da0d62cc5eeccbd7d5430efeefd045eac0038756d469e85546","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-b69b5201d4a2fd8f","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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-757de0468c6aa963","kind":"candidate_equal","node_a_id":"n-1a474575203b7460","node_b_id":"n-bd9b770a0539d41d","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-1a474575203b7460","other":{"id":"n-1a474575203b7460","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.","first_index":0,"fingerprint":"f6-8b732aeffa021ba5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","408","289"],["1728","7344","10404","4913"],["20736","117504","249696","235824","83521"]],"card_rows":[["144","408","289"],["1728","7344","10404","4913"],["20736","117504","249696","235824","83521"],["248832","1762560","4993920","7074720","5011260","1419857"]],"row_sums_preview":["841","24389","707281","20511149","594823321","17249876309","500246412961","14507145975869"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a5302200d285579d","steps":["x \u21a6 x/(x+1)"],"edges":["e-32f62515e143e962"],"text":"A038331 \u2192 x \u21a6 x/(x+1)"},"headline":"A038331 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-09caab113853dbce","kind":"fingerprint_related","node_a_id":"n-b69b5201d4a2fd8f","node_b_id":"n-bd9b770a0539d41d","fingerprint":"f6-8b732aeffa021ba5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8b732aeffa021ba5"},"other_id":"n-b69b5201d4a2fd8f","other":{"id":"n-b69b5201d4a2fd8f","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","first_index":0,"fingerprint":"f6-8b732aeffa021ba5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["289","408","144"],["4913","10404","7344","1728"],["83521","235824","249696","117504","20736"]],"card_rows":[["289","408","144"],["4913","10404","7344","1728"],["83521","235824","249696","117504","20736"],["1419857","5011260","7074720","4993920","1762560","248832"]],"row_sums_preview":["841","24389","707281","20511149","594823321","17249876309","500246412961","14507145975869"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f2dec04776afb2b","steps":["basis A013609"],"edges":["e-7304be3f66cc3caf"],"text":"A038320 \u2192 basis A013609"},"headline":"A038320 \u2192 basis A013609","chain":null}},{"id":"l-4005bcf94c4c9a1a","kind":"fingerprint_related","node_a_id":"n-5e42752da9a07882","node_b_id":"n-bd9b770a0539d41d","fingerprint":"f6-8b732aeffa021ba5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8b732aeffa021ba5"},"other_id":"n-5e42752da9a07882","other":{"id":"n-5e42752da9a07882","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","first_index":0,"fingerprint":"f6-8b732aeffa021ba5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["408","288"],["10404","14688","5184"],["235824","499392","352512","82944"]],"card_rows":[["408","288"],["10404","14688","5184"],["235824","499392","352512","82944"],["5011260","14149440","14981760","7050240","1244160"]],"row_sums_preview":["696","30276","1170672","42436860","1476802728","49965158964","1655988125664","54026612599788"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f2dec04776afb2b","steps":["basis A013609","D"],"edges":["e-7304be3f66cc3caf","e-e8f6022f8432a443"],"text":"A038320 \u2192 basis A013609 \u2192 D"},"headline":"A038320 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-6e1ecde078e10fdc","kind":"fingerprint_related","node_a_id":"n-29e4710712165c78","node_b_id":"n-bd9b770a0539d41d","fingerprint":"f6-8b732aeffa021ba5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8b732aeffa021ba5"},"other_id":"n-29e4710712165c78","other":{"id":"n-29e4710712165c78","name":"shift_x_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-8b732aeffa021ba5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["17","12"],["289","408","144"]],"card_rows":[["1"],["17","12"],["289","408","144"],["4913","10404","7344","1728"]],"row_sums_preview":["1","29","841","24389","707281","20511149","594823321","17249876309"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-1a3f71d8de8c31ec","e-037dd8eb46c1c7df"],"text":"A013620 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A013620 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-78a076821ef46272","kind":"fingerprint_related","node_a_id":"n-48ee1acaa798d2f7","node_b_id":"n-bd9b770a0539d41d","fingerprint":"f6-8b732aeffa021ba5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8b732aeffa021ba5"},"other_id":"n-48ee1acaa798d2f7","other":{"id":"n-48ee1acaa798d2f7","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-8b732aeffa021ba5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","17"],["144","408","289"]],"card_rows":[["1"],["12","17"],["144","408","289"],["1728","7344","10404","4913"]],"row_sums_preview":["1","29","841","24389","707281","20511149","594823321","17249876309"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-7ab50ded9e353f77","e-06ecdd435af05804"],"text":"A038271 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A038271 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-e12c6b1c3f80b745","kind":"fingerprint_related","node_a_id":"n-bd9b770a0539d41d","node_b_id":"n-6520f6faa376ffcc","fingerprint":"f6-8b732aeffa021ba5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8b732aeffa021ba5"},"other_id":"n-6520f6faa376ffcc","other":{"id":"n-6520f6faa376ffcc","name":"basis_map_A013609 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-8b732aeffa021ba5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1156","1632","576"],["39304","83232","58752","13824"],["1336336","3773184","3995136","1880064","331776"]],"card_rows":[["1156","1632","576"],["39304","83232","58752","13824"],["1336336","3773184","3995136","1880064","331776"],["45435424","160360320","226391040","159805440","56401920","7962624"]],"row_sums_preview":["3364","195112","11316496","656356768","38068692544","2207984167552","128063081718016","7427658739644928"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-e77df5b989acd52a","e-dc2ae8fdb5495b23"],"text":"A038314 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A038314 \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:          144,            408,            289;\n  1:         1728,           7344,          10404,            4913;\n  2:        20736,         117504,         249696,          235824,           83521;\n  3:       248832,        1762560,        4993920,         7074720,         5011260,          1419857;\n  4:      2985984,       25380864,       89890560,       169793280,       180405360,        102229704,         24137569;\n  5:     35831808,      355332096,     1510161408,      3565658880,      5051350080,       4293647568,       2027555796,        410338673;\n  6:    429981696,     4873125888,    24162582528,     68460650496,    121232401920,     137396722176,      97322678208,      39392512608,       6975757441;\n  7:   5159780352,    65787199488,   372794130432,   1232291708928,   2618619881472,    3709711498752,    3503616415488,    2127195680832,     753381803628,    118587876497;\n  8:  61917364224,   877162659840,  5591911956480,  21125000724480,  52372397629440,   89033075970048,  105108492464640,   85087827233280,   45202908217680,  14230545179640,   2015993900449;\n  9: 743008370688, 11578547109888, 82014708695040, 348562511953920, 987593783869440, 1958727671341056, 2774864201066496, 2807898298698240, 1988927961577920, 939215981856240, 266111194859268, 34271896307633;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12+17*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 408*x + 289*x^2.","mathematica":"P[0, x_] := 144 + 408*x + 289*x^2;\nP[n_, x_] := P[n, x] = Expand[(12 + 17*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] = 144 + 408*x + 289*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 17*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-1f2dec04776afb2b","steps":["basis A013609","reverse"],"edges":["e-7304be3f66cc3caf","e-dd8524b7a1597679"],"text":"A038320 \u2192 basis A013609 \u2192 reverse"},"headline":"A038320 \u2192 basis A013609 \u2192 reverse","chain":null}
