{"id":"n-79a292160b221b97","revision":0,"first_index":1,"fingerprint":"f6-800b3c5f1e1db21a","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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"],["2","6"],["8","48","56"],["32","288","768","576"],["128","1536","6400","10752","6016"],["512","7680","43520","115200","140800","62976"],["2048","36864","264192","958464","1843200","1769472","659456"],["8192","172032","1490944","6881280","18178048","27353088","21618688","6905856"],["32768","786432","7995392","44826624","151060480","312213504","385220608","258736128","72318976"],["131072","3538944","41287680","272498688","1118306304","2951479296","4996595712","5219942400","3048210432","757334016"],["524288","15728640","207093760","1572864000","7615283200","24508366848","52987166720","75843502080","68668620800","35468083200","7930904576"],["2097152","69206016","1015021568","8719958016","48674897920","185056886784","488156168192","891996340224","1104850976768","882376704000","408569249792","83053510656"],["8388608","301989888","4882169856","46808432640","296025587712","1299160498176","4051462782976","9033725509632","14275565322240","15574222503936","11124099514368","4667555708928","869747654656"],["33554432","1308622848","23119003648","244712472576","1729563197440","8609429716992","31016978743296","81791545245696","157696468713472","219331732439040","213709452476416","138041389744128","52952551129088","9108115685376"],["134217728","5637144576","108045271040","1251446095872","9776687742976","54443542315008","222644662173696","678583357931520","1547461013274624","2625144946163712","3258409554018304","2867491428630528","1690321289216000","597182184947712","95381425750016"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","112","1664","24832","370688","5533696","82608128","1233190912","18409324544","274818138112","4102541017088","61243566063616","914256400744448","13648205354893312"],"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":[["1"],["2","6"]],"recurrence":{"denominator":{"coeffs":[["-2","-4","-2"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["-8","-52","-100","-76","-20"],["4","26","50","38","10"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-16","-36","-32","-10"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185644,"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-dc49b3c9a81fbcd2"]},{"id":185645,"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-dc49b3c9a81fbcd2"]},{"id":185646,"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-dc49b3c9a81fbcd2"]}],"region_exact":{},"incoming_edges":[{"id":"e-dc49b3c9a81fbcd2","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-79a292160b221b97","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":"2b0ce6ceea51d124552056b2c7044809e2f5e2954f89b9572085a88bb9e62058","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-7ff58d52d83b38f4","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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-60cf7b5d25aa4ae9","kind":"fingerprint_related","node_a_id":"n-5c75d50b93f7f35c","node_b_id":"n-79a292160b221b97","fingerprint":"f6-800b3c5f1e1db21a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-800b3c5f1e1db21a"},"other_id":"n-5c75d50b93f7f35c","other":{"id":"n-5c75d50b93f7f35c","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","first_index":0,"fingerprint":"f6-800b3c5f1e1db21a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","6","7"]],"card_rows":[["1"],["1","3"],["1","6","7"],["1","9","24","18"]],"row_sums_preview":["1","4","14","52","194","724","2702","10084"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2496e2f555756362","steps":["x \u21a6 x/(x+1)"],"edges":["e-d394633f45845991"],"text":"A084534 \u2192 x \u21a6 x/(x+1)"},"headline":"A084534 \u2192 x \u21a6 x/(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:      1;\n   2:      2,       6;\n   3:      8,      48,       56;\n   4:     32,     288,      768,       576;\n   5:    128,    1536,     6400,     10752,       6016;\n   6:    512,    7680,    43520,    115200,     140800,      62976;\n   7:   2048,   36864,   264192,    958464,    1843200,    1769472,     659456;\n   8:   8192,  172032,  1490944,   6881280,   18178048,   27353088,   21618688,    6905856;\n   9:  32768,  786432,  7995392,  44826624,  151060480,  312213504,  385220608,  258736128,   72318976;\n  10: 131072, 3538944, 41287680, 272498688, 1118306304, 2951479296, 4996595712, 5219942400, 3048210432, 757334016;\n","recurrence_sentence":"The row polynomials satisfy (n-2+(2*n-4)*x+(n-2)*x^2)*P(n,x) = (4*n-8+(26*n-52)*x+(50*n-100)*x^2+(38*n-76)*x^3+(10*n-20)*x^4)*P(n-1,x) - (2*x+16*x^2+36*x^3+32*x^4+10*x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 6*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (26*n - 52)*x + (50*n - 100)*x^2 + (38*n - 76)*x^3 + (10*n - 20)*x^4)*P[n - 1, x] - (2*x + 16*x^2 + 36*x^3 + 32*x^4 + 10*x^5)*D[P[n - 1, x], x])/(n - 2 + (2*n - 4)*x + (n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 6*x; for(n = 3, N, my(i = n); v[i] = ((4*n - 8 + (26*n - 52)*x + (50*n - 100)*x^2 + (38*n - 76)*x^3 + (10*n - 20)*x^4)*v[i - 1] - (2*x + 16*x^2 + 36*x^3 + 32*x^4 + 10*x^5)*deriv(v[i - 1], x))/(n - 2 + (2*n - 4)*x + (n - 2)*x^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-f7935c36a3c0ec8c","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-11143fa0a471b62a","e-dc49b3c9a81fbcd2"],"text":"A086645 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A086645 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}
