{"id":"n-f2ad15cf169afade","revision":0,"first_index":0,"fingerprint":"f6-c115e0cff0f472ac","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","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"],["3","9"],["12","66","81"],["48","384","957","729"],["192","2016","7560","11892","6561"],["768","9984","49968","119712","136551","59049"],["3072","47616","297984","960384","1675284","1494678","531441"],["12288","221184","1661184","6731520","15855264","21654144","15843585","4782969"],["49152","1007616","8828928","43109376","128054400","236525760","264838224","164115624","43046721"],["196608","4521984","45281280","258736128","928360704","2166063360","3281913120","3109957440","1670750859","387420489"],["786432","20054016","225902592","1478688768","6221343744","17559876096","33635796480","43130229120","35399033820","16780149930","3486784401"],["3145728","88080384","1102381056","8133083136","39264731136","130124316672","301786700544","489384311040","543361208400","393121320960","166711879173","31381059609"],["12582912","383778816","5283250176","43375656960","236356632576","899836379136","2452397875200","4817365364736","6764565775680","6617925305760","4279469421912","1641621680796","282429536481"],["50331648","1660944384","24942477312","225542406144","1369261670400","5889714094080","18454572711936","42622321360896","72515836445184","89712900180480","78406879869648","45823089053664","16045528041327","2541865828329"],["201326592","7147094016","116266106880","1148140388352","7684921491456","36861092167680","130588222095360","346936485150720","694214978512896","1040653676891136","1149875429302272","907799827159296","483897777904740","155848148599422","22876792454961"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","159","2118","28221","376032","5010459","66762138","889575801","11853201972","157938645399","2104462217838","28041023241141","373634165415432","4978509106645779"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["0","0","0"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["12","67","145","153","79","16"],["-23","-123","-258","-263","-129","-24"],["14","71","134","119","50","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["6","59","208","362","338","163","32"],["-4","-38","-134","-230","-206","-92","-16"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","18","60","100","90","42","8"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184811,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-6b804da734008fb0"]},{"id":184812,"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-6b804da734008fb0"]},{"id":184813,"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-6b804da734008fb0"]}],"region_exact":{},"incoming_edges":[{"id":"e-6b804da734008fb0","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-f2ad15cf169afade","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":"47d2a1020b176ade65bea6c377411aae6786e3a10992ba9933cd93a890db5b77","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6aadffa116a44288","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","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":[],"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:      3,       9;\n  2:     12,      66,       81;\n  3:     48,     384,      957,       729;\n  4:    192,    2016,     7560,     11892,      6561;\n  5:    768,    9984,    49968,    119712,    136551,      59049;\n  6:   3072,   47616,   297984,    960384,   1675284,    1494678,     531441;\n  7:  12288,  221184,  1661184,   6731520,  15855264,   21654144,   15843585,    4782969;\n  8:  49152, 1007616,  8828928,  43109376, 128054400,  236525760,  264838224,  164115624,   43046721;\n  9: 196608, 4521984, 45281280, 258736128, 928360704, 2166063360, 3281913120, 3109957440, 1670750859, 387420489;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n^2*x+n^2*x^2)*P(n,x) = (14*n^2-23*n+12+(71*n^2-123*n+67)*x+(134*n^2-258*n+145)*x^2+(119*n^2-263*n+153)*x^3+(50*n^2-129*n+79)*x^4+(8*n^2-24*n+16)*x^5)*P(n-1,x) - (4*n-6+(38*n-59)*x+(134*n-208)*x^2+(230*n-362)*x^3+(206*n-338)*x^4+(92*n-163)*x^5+(16*n-32)*x^6)*d/dx P(n-1,x) + (2*x+18*x^2+60*x^3+100*x^4+90*x^5+42*x^6+8*x^7)*d^2/dx^2 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[((14*n^2 - 23*n + 12 + (71*n^2 - 123*n + 67)*x + (134*n^2 - 258*n + 145)*x^2 + (119*n^2 - 263*n + 153)*x^3 + (50*n^2 - 129*n + 79)*x^4 + (8*n^2 - 24*n + 16)*x^5)*P[n - 1, x] - (4*n - 6 + (38*n - 59)*x + (134*n - 208)*x^2 + (230*n - 362)*x^3 + (206*n - 338)*x^4 + (92*n - 163)*x^5 + (16*n - 32)*x^6)*D[P[n - 1, x], x] + (2*x + 18*x^2 + 60*x^3 + 100*x^4 + 90*x^5 + 42*x^6 + 8*x^7)*D[P[n - 1, x], {x, 2}])/(n^2 + 2*n^2*x + n^2*x^2)]];\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] = ((14*n^2 - 23*n + 12 + (71*n^2 - 123*n + 67)*x + (134*n^2 - 258*n + 145)*x^2 + (119*n^2 - 263*n + 153)*x^3 + (50*n^2 - 129*n + 79)*x^4 + (8*n^2 - 24*n + 16)*x^5)*v[i - 1] - (4*n - 6 + (38*n - 59)*x + (134*n - 208)*x^2 + (230*n - 362)*x^3 + (206*n - 338)*x^4 + (92*n - 163)*x^5 + (16*n - 32)*x^6)*deriv(v[i - 1], x) + (2*x + 18*x^2 + 60*x^3 + 100*x^4 + 90*x^5 + 42*x^6 + 8*x^7)*deriv(deriv(v[i - 1], x), x))/(n^2 + 2*n^2*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c85abc56078671cc","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-12f73d6f0ea24cf5","e-6b804da734008fb0"],"text":"A232535 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A232535 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}