{"id":"n-6130243e51f689b8","revision":0,"first_index":1,"fingerprint":"f6-bb0a136ce054ce63","name":"mobius_x_over_x_plus_one of Triangular array: T(n,k) = sqrt(C(n-1,k-1)*C(n-1,k)*C(n,k+1)* C(n+1,k+1)*C(n+1,k)*C(n,k-1)), where C(n,k) = binomial(n,k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["30","180","180"],["60","780","1980","1320"],["105","2520","12180","19320","9660"],["168","6720","54600","154560","177240","70896"],["252","15624","197820","887040","1750140","1567944","522648"],["360","32760","612360","4064760","12259800","18453960","13550040","3871440"],["495","63360","1677060","15744960","67976370","152792640","185377500","115239960","28809990"],["660","114840","4161960","53416440","316285200","994732200","1772555400","1798354800","968932800","215318400"],["858","197340","9523800","162625320","1280539260","5404679280","13257644400","19512979200","16995092400","8077898400","1615579680"],["1092","324324","20360340","452251800","4624139520","25429884480","82395593280","164915150400","206395469280","157382665440","66910203360","12165491520"],["1365","513240","41081040","1164443280","15163228080","106254308160","441792351000","1154057945040","1945588685040","2115675962400","1434095022360","551432634480","91905439080"],["1680","786240","78864240","2805522720","45773167440","401501580480","2096506011600","6946065846720","15140781255600","22011298589280","21145942818000","12897384601920","4526333862960","696359055840"],["2040","1170960","144984840","6379332960","128577048600","1391139750000","8969344183800","36900228736800","101082542160600","188582773819440","240713966012520","207011447086080","114739896518520","37032917848080","5290416835440"],["2448","1701360","256611600","13785997680","339020594640","4468036613040","35082390142800","176288738511600","594284828108400","1383062023783440","2251665303359472","2559674410609680","1991761431332400","1011517967733840","302182770918960","40291036122528"],["2907","2418624","439189560","28478832480","845105772420","13419517587840","126837841089960","768200482764000","3135611180357370","8915185502308800","17995869224941608","25959490444401696","26585991234309780","18884618160387840","8848548700032600","2460298442276592","307537305284574"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["390","4140","43785","464184","4941468","52845480","567682335","6123872700","66316759938","720691534836","7857171614565","85910831964720","941849775490680","10350632002143888","114002482061958651"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["30","180","180"]],"recurrence":{"denominator":{"coeffs":[["5","15","15","5"],["6","18","18","6"],["1","3","3","1"]]},"terms":[{"coeff":{"coeffs":[["20","100","180","140","40","0","0"],["9","39","45","-21","-78","-54","-12"],["1","11","45","91","98","54","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","27","132","282","348","267","120","24"],["0","3","-12","-102","-228","-237","-120","-24"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","6","42","120","180","150","66","12"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188892,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-54354f9285025db1"]},{"id":188893,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-54354f9285025db1"]},{"id":188894,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-54354f9285025db1"]}],"region_exact":{},"incoming_edges":[{"id":"e-54354f9285025db1","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":2,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-6130243e51f689b8","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"05845f7dd4e8867d014f285d2477c5fb5e06a4745f0c0a017fd643083d6e5092","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-05b91207a1ba3acf","role":"sequence","name":"Triangular array: T(n,k) = sqrt(C(n-1,k-1)*C(n-1,k)*C(n,k+1)* C(n+1,k+1)*C(n+1,k)*C(n,k-1)), where C(n,k) = binomial(n,k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-00e43a9e97502d9b","kind":"fingerprint_related","node_a_id":"n-6a437466bd6a8bc0","node_b_id":"n-6130243e51f689b8","fingerprint":"f6-bb0a136ce054ce63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bb0a136ce054ce63"},"other_id":"n-6a437466bd6a8bc0","other":{"id":"n-6a437466bd6a8bc0","name":"mobius_x_over_x_plus_one of Triangle read by rows. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 - n], [2, 3], -x) for n >= 1.","first_index":0,"fingerprint":"f6-bb0a136ce054ce63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","1","2"]],"card_rows":[["1"],["0","1"],["0","1","2"],["0","1","6","6"]],"row_sums_preview":["1","1","3","13","69","417","2763","19609"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-afb2f273dc5c03d1","steps":["x \u21a6 x/(x+1)"],"edges":["e-c5483277b29c066b"],"text":"A359363 \u2192 x \u21a6 x/(x+1)"},"headline":"A359363 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-0950582c8dc254d6","kind":"fingerprint_related","node_a_id":"n-d3f0e02b70bbc824","node_b_id":"n-6130243e51f689b8","fingerprint":"f6-bb0a136ce054ce63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bb0a136ce054ce63"},"other_id":"n-d3f0e02b70bbc824","other":{"id":"n-d3f0e02b70bbc824","name":"reversal of shift_x_plus_one of Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","first_index":0,"fingerprint":"f6-bb0a136ce054ce63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","13","33","22"]],"row_sums_preview":["1","3","13","69","417","2763","19609","146793"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-047228a38014e6d4","steps":["x \u21a6 x+1","reverse"],"edges":["e-0c759266bd791ca5","e-aa662e1e8e726553"],"text":"A056939 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A056939 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-55cfa08fc70437f2","kind":"fingerprint_related","node_a_id":"n-ed5623c8c46569fa","node_b_id":"n-6130243e51f689b8","fingerprint":"f6-bb0a136ce054ce63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bb0a136ce054ce63"},"other_id":"n-ed5623c8c46569fa","other":{"id":"n-ed5623c8c46569fa","name":"shift_x_plus_one of Triangular array: T(n,k) = sqrt(C(n-1,k-1)*C(n-1,k)*C(n,k+1)* C(n+1,k+1)*C(n+1,k)*C(n,k-1)), where C(n,k) = binomial(n,k).","first_index":1,"fingerprint":"f6-bb0a136ce054ce63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["180","180","30"],["1320","1980","780","60"],["9660","19320","12180","2520","105"]],"card_rows":[["180","180","30"],["1320","1980","780","60"],["9660","19320","12180","2520","105"],["70896","177240","154560","54600","6720","168"]],"row_sums_preview":["390","4140","43785","464184","4941468","52845480","567682335","6123872700"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-05b91207a1ba3acf","steps":["x \u21a6 x+1"],"edges":["e-edb392f7945e9505"],"text":"A197208 \u2192 x \u21a6 x+1"},"headline":"A197208 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-a351d0efcd5323e1","kind":"fingerprint_related","node_a_id":"n-33bd4b527bca505e","node_b_id":"n-6130243e51f689b8","fingerprint":"f6-bb0a136ce054ce63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bb0a136ce054ce63"},"other_id":"n-33bd4b527bca505e","other":{"id":"n-33bd4b527bca505e","name":"mobius_x_over_x_plus_one of Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","first_index":0,"fingerprint":"f6-bb0a136ce054ce63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","13","33","22"]],"row_sums_preview":["1","3","13","69","417","2763","19609","146793"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-047228a38014e6d4","steps":["x \u21a6 x/(x+1)"],"edges":["e-9da467f29ac457c6"],"text":"A056939 \u2192 x \u21a6 x/(x+1)"},"headline":"A056939 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-b389551e46951485","kind":"fingerprint_related","node_a_id":"n-563143c7a3f59683","node_b_id":"n-6130243e51f689b8","fingerprint":"f6-bb0a136ce054ce63","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bb0a136ce054ce63"},"other_id":"n-563143c7a3f59683","other":{"id":"n-563143c7a3f59683","name":"shift_x_plus_one of Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.","first_index":0,"fingerprint":"f6-bb0a136ce054ce63","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["6","6","1"]],"card_rows":[["1"],["2","1"],["6","6","1"],["22","33","13","1"]],"row_sums_preview":["1","3","13","69","417","2763","19609","146793"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-047228a38014e6d4","steps":["x \u21a6 x+1"],"edges":["e-0c759266bd791ca5"],"text":"A056939 \u2192 x \u21a6 x+1"},"headline":"A056939 \u2192 x \u21a6 x+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   30,    180,      180;\n   2:   60,    780,     1980,      1320;\n   3:  105,   2520,    12180,     19320,       9660;\n   4:  168,   6720,    54600,    154560,     177240,       70896;\n   5:  252,  15624,   197820,    887040,    1750140,     1567944,      522648;\n   6:  360,  32760,   612360,   4064760,   12259800,    18453960,    13550040,      3871440;\n   7:  495,  63360,  1677060,  15744960,   67976370,   152792640,   185377500,    115239960,     28809990;\n   8:  660, 114840,  4161960,  53416440,  316285200,   994732200,  1772555400,   1798354800,    968932800,    215318400;\n   9:  858, 197340,  9523800, 162625320, 1280539260,  5404679280, 13257644400,  19512979200,  16995092400,   8077898400,  1615579680;\n  10: 1092, 324324, 20360340, 452251800, 4624139520, 25429884480, 82395593280, 164915150400, 206395469280, 157382665440, 66910203360, 12165491520;\n","recurrence_sentence":"The row polynomials satisfy (n^2+6*n+5+(3*n^2+18*n+15)*x+(3*n^2+18*n+15)*x^2+(n^2+6*n+5)*x^3)*P(n,x) = (n^2+9*n+20+(11*n^2+39*n+100)*x+(45*n^2+45*n+180)*x^2+(91*n^2-21*n+140)*x^3+(98*n^2-78*n+40)*x^4+(54*n^2-54*n)*x^5+(12*n^2-12*n)*x^6)*P(n-1,x) + ((3*n+27)*x-(12*n-132)*x^2-(102*n-282)*x^3-(228*n-348)*x^4-(237*n-267)*x^5-(120*n-120)*x^6-(24*n-24)*x^7)*d/dx P(n-1,x) + (6*x^2+42*x^3+120*x^4+180*x^5+150*x^6+66*x^7+12*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 30 + 180*x + 180*x^2.","mathematica":"P[1, x_] := 30 + 180*x + 180*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 9*n + 20 + (11*n^2 + 39*n + 100)*x + (45*n^2 + 45*n + 180)*x^2 + (91*n^2 - 21*n + 140)*x^3 + (98*n^2 - 78*n + 40)*x^4 + (54*n^2 - 54*n)*x^5 + (12*n^2 - 12*n)*x^6)*P[n - 1, x] + ((3*n + 27)*x - (12*n - 132)*x^2 - (102*n - 282)*x^3 - (228*n - 348)*x^4 - (237*n - 267)*x^5 - (120*n - 120)*x^6 - (24*n - 24)*x^7)*D[P[n - 1, x], x] + (6*x^2 + 42*x^3 + 120*x^4 + 180*x^5 + 150*x^6 + 66*x^7 + 12*x^8)*D[P[n - 1, x], {x, 2}])/(n^2 + 6*n + 5 + (3*n^2 + 18*n + 15)*x + (3*n^2 + 18*n + 15)*x^2 + (n^2 + 6*n + 5)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 30 + 180*x + 180*x^2; for(n = 2, N, my(i = n); v[i] = ((n^2 + 9*n + 20 + (11*n^2 + 39*n + 100)*x + (45*n^2 + 45*n + 180)*x^2 + (91*n^2 - 21*n + 140)*x^3 + (98*n^2 - 78*n + 40)*x^4 + (54*n^2 - 54*n)*x^5 + (12*n^2 - 12*n)*x^6)*v[i - 1] + ((3*n + 27)*x - (12*n - 132)*x^2 - (102*n - 282)*x^3 - (228*n - 348)*x^4 - (237*n - 267)*x^5 - (120*n - 120)*x^6 - (24*n - 24)*x^7)*deriv(v[i - 1], x) + (6*x^2 + 42*x^3 + 120*x^4 + 180*x^5 + 150*x^6 + 66*x^7 + 12*x^8)*deriv(deriv(v[i - 1], x), x))/(n^2 + 6*n + 5 + (3*n^2 + 18*n + 15)*x + (3*n^2 + 18*n + 15)*x^2 + (n^2 + 6*n + 5)*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-05b91207a1ba3acf","steps":["x \u21a6 x/(x+1)"],"edges":["e-54354f9285025db1"],"text":"A197208 \u2192 x \u21a6 x/(x+1)"},"headline":"A197208 \u2192 x \u21a6 x/(x+1)","chain":null}