{"id":"n-8b9447648141fcb1","revision":0,"first_index":0,"fingerprint":"f6-2cb32fb53e87d6dc","name":"reversal of basis_map_A007318 of Even row Pascal-square read by antidiagonals.","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":false,"rows":[["1"],["2","3"],["3","10","8"],["4","22","38","21"],["5","40","111","130","55"],["6","65","256","474","420","144"],["7","98","511","1324","1836","1308","377"],["8","140","924","3130","6020","6666","3970","987"],["9","192","1554","6588","16435","25088","23109","11822","2584"],["10","255","2472","12720","39430","77645","98160","77378","34690","6765"],["11","330","3762","22968","85855","209034","339535","366108","252177","100610","17711"],["12","418","5522","39303","173272","506408","1013292","1399478","1315072","804228","289032","46368"],["13","520","7865","64350","328900","1128920","2702350","4584620","5504650","4582600","2519640","823800","121393"],["14","637","10920","101530","593450","2352285","6592080","13343820","19635840","20845300","15574520","7777860","2332418","317811"],["15","770","14833","155220","1026025","4633850","14951925","35345640","61926900","80421600","76495450","51833600","23709783","6566290","832040"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","21","85","341","1365","5461","21845","87381","349525","1398101","5592405","22369621","89478485","357913941"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"]]},"terms":[{"coeff":{"coeffs":[["2","7","5"],["2","-1","-5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","9","5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176374,"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-b49b02e17cf906b8"]},{"id":176375,"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-b49b02e17cf906b8"]},{"id":176376,"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-b49b02e17cf906b8"]}],"region_exact":{},"incoming_edges":[{"id":"e-b49b02e17cf906b8","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-8b9447648141fcb1","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":"0f5963355be0598952f13f318ff92415a3e48028475ac59c2236899d1c982408","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-74b50e0ea94d74b6","role":"sequence","name":"basis_map_A007318 of Even row Pascal-square read by antidiagonals.","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-0902f0c149d02da9","kind":"fingerprint_related","node_a_id":"n-8b9447648141fcb1","node_b_id":"n-93c87bd66e848d3f","fingerprint":"f6-2cb32fb53e87d6dc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2cb32fb53e87d6dc"},"other_id":"n-93c87bd66e848d3f","other":{"id":"n-93c87bd66e848d3f","name":"derivative of basis_map_A007318 of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).","first_index":0,"fingerprint":"f6-2cb32fb53e87d6dc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["3"],["15","10"]],"card_rows":[["3"],["15","10"],["56","70","21"],["189","342","198","36"]],"row_sums_preview":["0","3","25","147","765","3751","17745","81915"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9496336e2b29e28e","steps":["x \u21a6 x+1","D"],"edges":["e-557cf165357d4e1f","e-7f8f05cc7688eff8"],"text":"A082985 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A082985 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-2553fcf92fc6178e","kind":"fingerprint_related","node_a_id":"n-74b50e0ea94d74b6","node_b_id":"n-8b9447648141fcb1","fingerprint":"f6-2cb32fb53e87d6dc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2cb32fb53e87d6dc"},"other_id":"n-74b50e0ea94d74b6","other":{"id":"n-74b50e0ea94d74b6","name":"basis_map_A007318 of Even row Pascal-square read by antidiagonals.","first_index":0,"fingerprint":"f6-2cb32fb53e87d6dc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["8","10","3"]],"card_rows":[["1"],["3","2"],["8","10","3"],["21","38","22","4"]],"row_sums_preview":["1","5","21","85","341","1365","5461","21845"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79ebafb266da3b41","steps":["basis A007318"],"edges":["e-1a8887aa0d003bc9"],"text":"A172431 \u2192 basis A007318"},"headline":"A172431 \u2192 basis A007318","chain":null}},{"id":"l-25c0558d336271a7","kind":"fingerprint_related","node_a_id":"n-8b9447648141fcb1","node_b_id":"n-a10f777f75d58fc1","fingerprint":"f6-2cb32fb53e87d6dc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2cb32fb53e87d6dc"},"other_id":"n-a10f777f75d58fc1","other":{"id":"n-a10f777f75d58fc1","name":"shift_x_plus_one of basis_map_A013611 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","first_index":0,"fingerprint":"f6-2cb32fb53e87d6dc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","8"],["128","160","48"]],"card_rows":[["1"],["12","8"],["128","160","48"],["1344","2432","1408","256"]],"row_sums_preview":["1","20","336","5440","87296","1397760","22368256","357908480"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2245b35ec7cd4857","steps":["basis A013611","basis A007318"],"edges":["e-61d2480a3633cd97","e-3f0f6b82c3a39a27"],"text":"A091044 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A091044 \u2192 basis A013611 \u2192 basis A007318","chain":null}},{"id":"l-617742c305781456","kind":"fingerprint_related","node_a_id":"n-4d6b9eb76ece53c0","node_b_id":"n-8b9447648141fcb1","fingerprint":"f6-2cb32fb53e87d6dc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2cb32fb53e87d6dc"},"other_id":"n-4d6b9eb76ece53c0","other":{"id":"n-4d6b9eb76ece53c0","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","first_index":1,"fingerprint":"f6-2cb32fb53e87d6dc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["3","10","8"]],"card_rows":[["1"],["2","3"],["3","10","8"],["4","22","38","21"]],"row_sums_preview":["1","5","21","85","341","1365","5461","21845"],"regions":["[-1, 0)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-976cdb6e4a693ec8","steps":["x \u21a6 x/(x+1)"],"edges":["e-458bf7ea3df11c85"],"text":"A078812 \u2192 x \u21a6 x/(x+1)"},"headline":"A078812 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-e4b219c2f3880eae","kind":"fingerprint_related","node_a_id":"n-91f8c3beb9aeb119","node_b_id":"n-8b9447648141fcb1","fingerprint":"f6-2cb32fb53e87d6dc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2cb32fb53e87d6dc"},"other_id":"n-91f8c3beb9aeb119","other":{"id":"n-91f8c3beb9aeb119","name":"shift_x_plus_one of reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","first_index":1,"fingerprint":"f6-2cb32fb53e87d6dc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["8","10","3"]],"card_rows":[["1"],["3","2"],["8","10","3"],["21","38","22","4"]],"row_sums_preview":["1","5","21","85","341","1365","5461","21845"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-976cdb6e4a693ec8","steps":["reverse","x \u21a6 x+1"],"edges":["e-32f4efbf40925a37","e-2bd92cd6cc3ac96b"],"text":"A078812 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A078812 \u2192 reverse \u2192 x \u21a6 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:  1;\n  1:  2,   3;\n  2:  3,  10,    8;\n  3:  4,  22,   38,    21;\n  4:  5,  40,  111,   130,    55;\n  5:  6,  65,  256,   474,   420,   144;\n  6:  7,  98,  511,  1324,  1836,  1308,   377;\n  7:  8, 140,  924,  3130,  6020,  6666,  3970,   987;\n  8:  9, 192, 1554,  6588, 16435, 25088, 23109, 11822,  2584;\n  9: 10, 255, 2472, 12720, 39430, 77645, 98160, 77378, 34690, 6765;\n","recurrence_sentence":"The row polynomials satisfy 2*n*P(n,x) = (2*n+2-(n-7)*x-(5*n-5)*x^2)*P(n-1,x) + (4*x+9*x^2+5*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 - (n - 7)*x - (5*n - 5)*x^2)*P[n - 1, x] + (4*x + 9*x^2 + 5*x^3)*D[P[n - 1, x], x])/(2*n)]];\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] = ((2*n + 2 - (n - 7)*x - (5*n - 5)*x^2)*v[i - 1] + (4*x + 9*x^2 + 5*x^3)*deriv(v[i - 1], x))/(2*n)); 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-79ebafb266da3b41","steps":["basis A007318","reverse"],"edges":["e-1a8887aa0d003bc9","e-b49b02e17cf906b8"],"text":"A172431 \u2192 basis A007318 \u2192 reverse"},"headline":"A172431 \u2192 basis A007318 \u2192 reverse","chain":null}
