{"id":"n-830d062cddf4b58c","revision":0,"first_index":0,"fingerprint":"f6-4f0d71d8efeb5cf0","name":"reversal of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^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":[["9","66","121"],["27","297","1089","1331"],["81","1188","6534","15972","14641"],["243","4455","32670","119790","219615","161051"],["729","16038","147015","718740","1976535","2898918","1771561"],["2187","56133","617463","3773385","13835745","30438639","37202781","19487171"],["6561","192456","2469852","18112248","83014470","243509112","446433372","467692104","214358881"],["19683","649539","9526572","81505116","448278138","1643686506","4017900348","6313843404","5787689787","2357947691"],["59049","2165130","35724645","349307640","2241390690","9862119036","30134252610","63138434040","86815346805","70738430730","25937424601"],["177147","7144929","130990365","1440894015","10566556110","54241654698","198886067226","520892080830","954968814855","1167184107045","855935011833","285311670611"],["531441","23383404","471565314","5763576060","47549502495","278957081304","1193316403356","3750422981976","8594719333695","14006209284540","15406830212994","10271220141996","3138428376721"],["1594323","75996063","1671913386","22477946634","206047844145","1359915771357","6648477104412","24377749382844","67038810802821","136560540524265","200288792768922","200288792768922","122398706692119","34522712143931"],["4782969","245525742","5851696851","85824887148","865400945409","6346273599666","34904504798163","146266496297064","469271675619747","1147108540403826","2103032324073681","2804043098764908","2570372840534499","1449953910045102","379749833583241"],["14348907","789189885","20255873715","321843326805","3540276594855","28558231198497","174522523990815","822749041670985","3016746486126945","8603314053028695","18927290916663129","31545484861105215","38555592608017485","32623962976014795","17088742511245845","4177248169415651"],["43046721","2525407632","69448709880","1188344591280","14161106379420","124617736138896","837708115155912","4387994888911920","18100478916761670","58994153506482480","151418327333305032","302836654666610064","462667111296209820","521983407616236720","410129820269900280","200507912131951248","45949729863572161"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["196","2744","38416","537824","7529536","105413504","1475789056","20661046784","289254654976","4049565169664","56693912375296","793714773254144","11112006825558016","155568095557812224","2177953337809371136"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["9","66","121"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","11"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192008,"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-4925a713d49e089b"]},{"id":192009,"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-4925a713d49e089b"]},{"id":192010,"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-4925a713d49e089b"]}],"region_exact":{},"incoming_edges":[{"id":"e-4925a713d49e089b","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-830d062cddf4b58c","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":"35c2b43f51b2265b0dd70ea60c364ce4e50df5a8829fe721c6913e72a0b41238","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-809ca598570cd420","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^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-ee8d3a7b361984bf","kind":"candidate_equal","node_a_id":"n-45494a7331411e11","node_b_id":"n-830d062cddf4b58c","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-45494a7331411e11","other":{"id":"n-45494a7331411e11","name":"mobius_x_over_x_plus_one of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","first_index":0,"fingerprint":"f6-4f0d71d8efeb5cf0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","66","121"],["27","297","1089","1331"],["81","1188","6534","15972","14641"]],"card_rows":[["9","66","121"],["27","297","1089","1331"],["81","1188","6534","15972","14641"],["243","4455","32670","119790","219615","161051"]],"row_sums_preview":["196","2744","38416","537824","7529536","105413504","1475789056","20661046784"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-26633b980c3508e8","steps":["x \u21a6 x/(x+1)"],"edges":["e-6c857c2ccbfe6a6e"],"text":"A038226 \u2192 x \u21a6 x/(x+1)"},"headline":"A038226 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-0b973cb917c52d79","kind":"fingerprint_related","node_a_id":"n-bfe674a999ae5db6","node_b_id":"n-830d062cddf4b58c","fingerprint":"f6-4f0d71d8efeb5cf0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f0d71d8efeb5cf0"},"other_id":"n-bfe674a999ae5db6","other":{"id":"n-bfe674a999ae5db6","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","first_index":0,"fingerprint":"f6-4f0d71d8efeb5cf0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["66","18"],["1089","594","81"],["15972","13068","3564","324"]],"card_rows":[["66","18"],["1089","594","81"],["15972","13068","3564","324"],["219615","239580","98010","17820","1215"]],"row_sums_preview":["84","1764","32928","576240","9680832","158120256","2529924096","39846304512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-108b34f10b7468f4","steps":["x \u21a6 x+1","D"],"edges":["e-3a9169eda66e0655","e-3ba08d9a3f5ce34f"],"text":"A038281 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A038281 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-23e857ae583f48ba","kind":"fingerprint_related","node_a_id":"n-809ca598570cd420","node_b_id":"n-830d062cddf4b58c","fingerprint":"f6-4f0d71d8efeb5cf0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f0d71d8efeb5cf0"},"other_id":"n-809ca598570cd420","other":{"id":"n-809ca598570cd420","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","first_index":0,"fingerprint":"f6-4f0d71d8efeb5cf0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["121","66","9"],["1331","1089","297","27"],["14641","15972","6534","1188","81"]],"card_rows":[["121","66","9"],["1331","1089","297","27"],["14641","15972","6534","1188","81"],["161051","219615","119790","32670","4455","243"]],"row_sums_preview":["196","2744","38416","537824","7529536","105413504","1475789056","20661046784"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-108b34f10b7468f4","steps":["x \u21a6 x+1"],"edges":["e-3a9169eda66e0655"],"text":"A038281 \u2192 x \u21a6 x+1"},"headline":"A038281 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-675062817fedd096","kind":"fingerprint_related","node_a_id":"n-314866308b34ea0d","node_b_id":"n-830d062cddf4b58c","fingerprint":"f6-4f0d71d8efeb5cf0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f0d71d8efeb5cf0"},"other_id":"n-314866308b34ea0d","other":{"id":"n-314866308b34ea0d","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+11x)^n.","first_index":0,"fingerprint":"f6-4f0d71d8efeb5cf0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","1056","1936"],["1728","19008","69696","85184"],["20736","304128","1672704","4088832","3748096"]],"card_rows":[["144","1056","1936"],["1728","19008","69696","85184"],["20736","304128","1672704","4088832","3748096"],["248832","4561920","33454080","122664960","224885760","164916224"]],"row_sums_preview":["3136","175616","9834496","550731776","30840979456","1727094849536","96717311574016","5416169448144896"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1e393c17b0898c2","steps":["basis A013611"],"edges":["e-f283d143fa0950f0"],"text":"A013618 \u2192 basis A013611"},"headline":"A013618 \u2192 basis A013611","chain":null}},{"id":"l-6d55fbdbaee79f63","kind":"fingerprint_related","node_a_id":"n-2de16a679fd7052d","node_b_id":"n-830d062cddf4b58c","fingerprint":"f6-4f0d71d8efeb5cf0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f0d71d8efeb5cf0"},"other_id":"n-2de16a679fd7052d","other":{"id":"n-2de16a679fd7052d","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","first_index":0,"fingerprint":"f6-4f0d71d8efeb5cf0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["66","18"]],"card_rows":[["3"],["66","18"],["1089","594","81"],["15972","13068","3564","324"]],"row_sums_preview":["0","3","84","1764","32928","576240","9680832","158120256"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013610","D"],"edges":["e-254f1b828fe6620e","e-ecf9960f88134af1"],"text":"A038303 \u2192 basis A013610 \u2192 D"},"headline":"A038303 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-801eb3157e37c4c5","kind":"fingerprint_related","node_a_id":"n-3832f5f734db009a","node_b_id":"n-830d062cddf4b58c","fingerprint":"f6-4f0d71d8efeb5cf0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f0d71d8efeb5cf0"},"other_id":"n-3832f5f734db009a","other":{"id":"n-3832f5f734db009a","name":"reversal of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","first_index":0,"fingerprint":"f6-4f0d71d8efeb5cf0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","11"],["9","66","121"]],"card_rows":[["1"],["3","11"],["9","66","121"],["27","297","1089","1331"]],"row_sums_preview":["1","14","196","2744","38416","537824","7529536","105413504"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013610","reverse"],"edges":["e-254f1b828fe6620e","e-23fe265bb0c329ed"],"text":"A038303 \u2192 basis A013610 \u2192 reverse"},"headline":"A038303 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-eeca92008cd9beea","kind":"fingerprint_related","node_a_id":"n-3c11bfaa4e68c9bc","node_b_id":"n-830d062cddf4b58c","fingerprint":"f6-4f0d71d8efeb5cf0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f0d71d8efeb5cf0"},"other_id":"n-3c11bfaa4e68c9bc","other":{"id":"n-3c11bfaa4e68c9bc","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.","first_index":0,"fingerprint":"f6-4f0d71d8efeb5cf0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","3"],["121","66","9"]],"card_rows":[["1"],["11","3"],["121","66","9"],["1331","1089","297","27"]],"row_sums_preview":["1","14","196","2744","38416","537824","7529536","105413504"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a920b56c9b3ef4d1","steps":["basis A013610"],"edges":["e-254f1b828fe6620e"],"text":"A038303 \u2192 basis A013610"},"headline":"A038303 \u2192 basis A013610","chain":null}},{"id":"l-f3380f6500bb9081","kind":"fingerprint_related","node_a_id":"n-255625afc7c906eb","node_b_id":"n-830d062cddf4b58c","fingerprint":"f6-4f0d71d8efeb5cf0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4f0d71d8efeb5cf0"},"other_id":"n-255625afc7c906eb","other":{"id":"n-255625afc7c906eb","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-4f0d71d8efeb5cf0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","11"],["9","66","121"]],"card_rows":[["1"],["3","11"],["9","66","121"],["27","297","1089","1331"]],"row_sums_preview":["1","14","196","2744","38416","537824","7529536","105413504"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["x \u21a6 x/(x+1)","x \u21a6 x/(x+1)"],"edges":["e-22d203db774fd047","e-a2c60af61ae08b44"],"text":"A013622 \u2192 x \u21a6 x/(x+1) \u2192 x \u21a6 x/(x+1)"},"headline":"A013622 \u2192 x \u21a6 x/(x+1) \u2192 x \u21a6 x/(x+1)","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:      9,      66,       121;\n  1:     27,     297,      1089,       1331;\n  2:     81,    1188,      6534,      15972,       14641;\n  3:    243,    4455,     32670,     119790,      219615,      161051;\n  4:    729,   16038,    147015,     718740,     1976535,     2898918,      1771561;\n  5:   2187,   56133,    617463,    3773385,    13835745,    30438639,     37202781,     19487171;\n  6:   6561,  192456,   2469852,   18112248,    83014470,   243509112,    446433372,    467692104,    214358881;\n  7:  19683,  649539,   9526572,   81505116,   448278138,  1643686506,   4017900348,   6313843404,   5787689787,    2357947691;\n  8:  59049, 2165130,  35724645,  349307640,  2241390690,  9862119036,  30134252610,  63138434040,  86815346805,   70738430730,  25937424601;\n  9: 177147, 7144929, 130990365, 1440894015, 10566556110, 54241654698, 198886067226, 520892080830, 954968814855, 1167184107045, 855935011833, 285311670611;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 9 + 66*x + 121*x^2.","mathematica":"P[0, x_] := 9 + 66*x + 121*x^2;\nP[n_, x_] := P[n, x] = Expand[(3 + 11*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] = 9 + 66*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 11*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-108b34f10b7468f4","steps":["x \u21a6 x+1","reverse"],"edges":["e-3a9169eda66e0655","e-4925a713d49e089b"],"text":"A038281 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A038281 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}