{"id":"n-c9014249fe74edd2","revision":0,"first_index":0,"fingerprint":"f6-b24c47ab26a256b2","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k peaks of the form Ud.","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","1"],["10","9","2"],["66","85","36","5"],["498","833","516","140","14"],["4066","8361","6800","2730","540","42"],["34970","85305","85830","45540","13420","2079","132"],["312066","880685","1055460","695695","272090","63063","8008","429"],["2862562","9173505","12754952","10042760","4893420","1509508","287560","30888","1430"],["26824386","96220561","152235072","139351380","81276048","31298904","7951104","1283568","119340","4862"],["255680170","1014889769","1800137610","1877960760","1275363600","588777966","186993744","40310400","5639580","461890","16796"],["2471150402","10753517061","21132768580","24746257305","19176377220","10320320934","3933808032","1061316630","198470580","24480170","1790712","58786"],["24161357010","114375683009","246660518412","320330935540","278900250090","171433444644","76246522968","24709864344","5787724140","954886515","105254072","6953544","208012"],["238552980386","1220435354425","2865465382416","4086998065150","3949525079500","2729734657650","1387679092800","525111873000","147840714450","30569034925","4509350560","449083050","27041560","742900"],["2375085745978","13058529727833","33157921067726","51523199336700","54719259118524","42003568131525","24024495795300","10397268367200","3419917493850","850410335775","157288761630","20970729780","1904043960","105306075","2674440"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","21","192","2001","22539","267276","3287496","41556585","536565225","7046232285","93820316412","1263673602300","17186898452772","235709926636296"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"],["4"]]},"terms":[{"coeff":{"coeffs":[["12","7","1"],["-52","-27","-2"],["52","26","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","14","7","1"],["-20","-32","-13","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180867,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-51315148ff798987"]},{"id":180868,"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-51315148ff798987"]},{"id":180869,"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-51315148ff798987"]}],"region_exact":{},"incoming_edges":[{"id":"e-51315148ff798987","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-c9014249fe74edd2","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":"1e2f0a9531a4a22cd729d0bc6e37f14c4856d0a1d80c836f197281bae0f079f6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-8e6b51ac707373f6","role":"sequence","name":"Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k peaks of the form Ud.","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-65403b0832646e27","kind":"fingerprint_related","node_a_id":"n-c9014249fe74edd2","node_b_id":"n-15250cda9c30cc03","fingerprint":"f6-b24c47ab26a256b2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b24c47ab26a256b2"},"other_id":"n-15250cda9c30cc03","other":{"id":"n-15250cda9c30cc03","name":"mobius_x_over_x_plus_one of Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.","first_index":0,"fingerprint":"f6-b24c47ab26a256b2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["2","9","10"]],"card_rows":[["1"],["1","2"],["2","9","10"],["5","36","85","66"]],"row_sums_preview":["1","3","21","192","2001","22539","267276","3287496"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9ed87ca196739b1c","steps":["x \u21a6 x/(x+1)"],"edges":["e-1d7bbf54cf140861"],"text":"A104978 \u2192 x \u21a6 x/(x+1)"},"headline":"A104978 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f14d7843802c7c36","kind":"fingerprint_related","node_a_id":"n-c9014249fe74edd2","node_b_id":"n-b11137933d9f04e2","fingerprint":"f6-b24c47ab26a256b2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b24c47ab26a256b2"},"other_id":"n-b11137933d9f04e2","other":{"id":"n-b11137933d9f04e2","name":"mobius_x_over_x_plus_one of Triangle read by rows, interpolating between the central binomial coefficients and the central coefficients of the Catalan triangle. T(n, k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-b24c47ab26a256b2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["6","27","30"]],"card_rows":[["1"],["2","4"],["6","27","30"],["20","144","340","264"]],"row_sums_preview":["1","6","63","768","10005","135234","1870932","26299968"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-eb78d61c7c9e8c34","steps":["x \u21a6 x/(x+1)"],"edges":["e-b7c2223d93149e78"],"text":"A330798 \u2192 x \u21a6 x/(x+1)"},"headline":"A330798 \u2192 x \u21a6 x/(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,        1;\n  2:       10,        9,         2;\n  3:       66,       85,        36,         5;\n  4:      498,      833,       516,       140,       14;\n  5:     4066,     8361,      6800,      2730,      540,       42;\n  6:    34970,    85305,     85830,     45540,    13420,     2079,     132;\n  7:   312066,   880685,   1055460,    695695,   272090,    63063,    8008,     429;\n  8:  2862562,  9173505,  12754952,  10042760,  4893420,  1509508,  287560,   30888,   1430;\n  9: 26824386, 96220561, 152235072, 139351380, 81276048, 31298904, 7951104, 1283568, 119340, 4862;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+2*n)*P(n,x) = (52*n^2-52*n+12+(26*n^2-27*n+7)*x+(n^2-2*n+1)*x^2)*P(n-1,x) - (20*n-8+(32*n-14)*x+(13*n-7)*x^2+(n-1)*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[((52*n^2 - 52*n + 12 + (26*n^2 - 27*n + 7)*x + (n^2 - 2*n + 1)*x^2)*P[n - 1, x] - (20*n - 8 + (32*n - 14)*x + (13*n - 7)*x^2 + (n - 1)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 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] = ((52*n^2 - 52*n + 12 + (26*n^2 - 27*n + 7)*x + (n^2 - 2*n + 1)*x^2)*v[i - 1] - (20*n - 8 + (32*n - 14)*x + (13*n - 7)*x^2 + (n - 1)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 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-8e6b51ac707373f6","steps":["x \u21a6 x+1"],"edges":["e-51315148ff798987"],"text":"A108426 \u2192 x \u21a6 x+1"},"headline":"A108426 \u2192 x \u21a6 x+1","chain":null}