{"id":"n-91f8c3beb9aeb119","revision":0,"first_index":1,"fingerprint":"f6-2cb32fb53e87d6dc","name":"shift_x_plus_one of reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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":[["1"],["3","2"],["8","10","3"],["21","38","22","4"],["55","130","111","40","5"],["144","420","474","256","65","6"],["377","1308","1836","1324","511","98","7"],["987","3970","6666","6020","3130","924","140","8"],["2584","11822","23109","25088","16435","6588","1554","192","9"],["6765","34690","77378","98160","77645","39430","12720","2472","255","10"],["17711","100610","252177","366108","339535","209034","85855","22968","3762","330","11"],["46368","289032","804228","1315072","1399478","1013292","506408","173272","39303","5522","418","12"],["121393","823800","2519640","4582600","5504650","4584620","2702350","1128920","328900","64350","7865","520","13"],["317811","2332418","7777860","15574520","20845300","19635840","13343820","6592080","2352285","593450","101530","10920","637","14"],["832040","6566290","23709783","51833600","76495450","80421600","61926900","35345640","14951925","4633850","1026025","155220","14833","770","15"]],"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:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["2"]]},"terms":[{"coeff":{"coeffs":[["-10","-8"],["8","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-5","-9","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172902,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-2bd92cd6cc3ac96b"]},{"id":172903,"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-2bd92cd6cc3ac96b"]},{"id":172904,"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-2bd92cd6cc3ac96b"]}],"region_exact":{},"incoming_edges":[{"id":"e-2bd92cd6cc3ac96b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-91f8c3beb9aeb119","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"3ed250e5ca0e26d0748c1dbc839c9f595b3d059e8e9004fb5a4b9f86bd414a34","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-697c605ecd7e51b4","role":"sequence","name":"reversal of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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-05f30f3788f63071","kind":"fingerprint_related","node_a_id":"n-4d6b9eb76ece53c0","node_b_id":"n-91f8c3beb9aeb119","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-27838e944edaaed9","kind":"fingerprint_related","node_a_id":"n-74b50e0ea94d74b6","node_b_id":"n-91f8c3beb9aeb119","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-ce3a38b6ed8074cc","kind":"fingerprint_related","node_a_id":"n-91f8c3beb9aeb119","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-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-8b9447648141fcb1","other":{"id":"n-8b9447648141fcb1","name":"reversal of 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"],["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":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-ff803cee5397eb26","kind":"fingerprint_related","node_a_id":"n-91f8c3beb9aeb119","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}}],"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:    1;\n   2:    3,     2;\n   3:    8,    10,     3;\n   4:   21,    38,    22,     4;\n   5:   55,   130,   111,    40,     5;\n   6:  144,   420,   474,   256,    65,     6;\n   7:  377,  1308,  1836,  1324,   511,    98,     7;\n   8:  987,  3970,  6666,  6020,  3130,   924,   140,    8;\n   9: 2584, 11822, 23109, 25088, 16435,  6588,  1554,  192,   9;\n  10: 6765, 34690, 77378, 98160, 77645, 39430, 12720, 2472, 255, 10;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = (8*n-10+(6*n-8)*x)*P(n-1,x) - (5+9*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n - 10 + (6*n - 8)*x)*P[n - 1, x] - (5 + 9*x + 4*x^2)*D[P[n - 1, x], x])/(2*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n - 10 + (6*n - 8)*x)*v[i - 1] - (5 + 9*x + 4*x^2)*deriv(v[i - 1], x))/(2*n - 2)); 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-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}