{"id":"n-51355abba27f4384","revision":0,"first_index":0,"fingerprint":"f6-b19b142556626fc3","name":"shift_x_plus_one of reversal of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["6","5","1"],["18","21","8","1"],["36","60","37","10","1"],["108","216","171","67","13","1"],["216","540","558","305","93","15","1"],["648","1836","2214","1473","584","138","18","1"],["1296","4320","6264","5160","2641","860","174","20","1"],["3888","14256","23112","21744","13083","5221","1382","234","23","1"],["7776","32400","60480","66600","47910","23525","7985","1850","280","25","1"],["23328","104976","213840","260280","210330","118485","47480","13535","2690","355","28","1"],["46656","233280","532656","734400","680940","447300","213445","74550","18915","3400","411","30","1"],["139968","746496","1831248","2735856","2777220","2022840","1087635","437095","131295","29115","4633","501","33","1"],["279936","1632960","4408992","7302960","8290296","6822900","4198110","1961825","699685","189525","38381","5635","567","35","1"],["839808","5178816","14859936","26317872","32173848","28758996","19417230","10083585","4060880","1268260","304668","55286","7336","672","38","1"],["1679616","11197440","34898688","67495680","90665568","89691840","67593456","39584400","18205345","6597400","1877596","415240","69958","8680","748","40","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["12","48","144","576","1728","6912","20736","82944","248832","995328","2985984","11943936","35831808","143327232","429981696"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["6","5","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2","-1"],["-5","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["12","10","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173377,"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-f87cc16a200c95f5"]},{"id":173378,"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-f87cc16a200c95f5"]},{"id":173379,"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-f87cc16a200c95f5"]}],"region_exact":{},"incoming_edges":[{"id":"e-f87cc16a200c95f5","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-51355abba27f4384","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":"c728636f49ca584a29053cc4174665419349e209b68448d1969e58f79275ffa8","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-1453c6ad660cd835","role":"sequence","name":"reversal of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","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-36bffeb9afef2cd3","kind":"fingerprint_related","node_a_id":"n-51355abba27f4384","node_b_id":"n-b8674991adb819c5","fingerprint":"f6-b19b142556626fc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b19b142556626fc3"},"other_id":"n-b8674991adb819c5","other":{"id":"n-b8674991adb819c5","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle read by rows: triangle of repeated stepped binomial coefficients.","first_index":0,"fingerprint":"f6-b19b142556626fc3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","3"],["2","10","12"]],"card_rows":[["1"],["1","3"],["2","10","12"],["2","16","42","36"]],"row_sums_preview":["1","4","24","96","576","2304","13824","55296"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-35bd700768cac55f","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-69752228c1d1e91e","e-34aab2eb112dbe5d"],"text":"A103633 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A103633 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-9f3e906aacc2ac73","kind":"fingerprint_related","node_a_id":"n-899fb13ecc891703","node_b_id":"n-51355abba27f4384","fingerprint":"f6-b19b142556626fc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b19b142556626fc3"},"other_id":"n-899fb13ecc891703","other":{"id":"n-899fb13ecc891703","name":"basis_map_A013611 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","first_index":0,"fingerprint":"f6-b19b142556626fc3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","12"],["8","40","48"]],"card_rows":[["1"],["4","12"],["8","40","48"],["32","256","672","576"]],"row_sums_preview":["1","16","96","1536","9216","147456","884736","14155776"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7196be931c2ed996","steps":["basis A013611"],"edges":["e-541f0d63dcdb93da"],"text":"A152842 \u2192 basis A013611"},"headline":"A152842 \u2192 basis A013611","chain":null}},{"id":"l-b5b5e49e11672905","kind":"fingerprint_related","node_a_id":"n-5066d86ba8adc739","node_b_id":"n-51355abba27f4384","fingerprint":"f6-b19b142556626fc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b19b142556626fc3"},"other_id":"n-5066d86ba8adc739","other":{"id":"n-5066d86ba8adc739","name":"mobius_x_over_x_plus_one of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","first_index":0,"fingerprint":"f6-b19b142556626fc3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","5","6"],["1","8","21","18"],["1","10","37","60","36"]],"card_rows":[["1","5","6"],["1","8","21","18"],["1","10","37","60","36"],["1","13","67","171","216","108"]],"row_sums_preview":["12","48","144","576","1728","6912","20736","82944"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5a70a6e97131d437","steps":["x \u21a6 x/(x+1)"],"edges":["e-5c2afa55aadc0623"],"text":"A064861 \u2192 x \u21a6 x/(x+1)"},"headline":"A064861 \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:     6,      5,      1;\n  1:    18,     21,      8,      1;\n  2:    36,     60,     37,     10,      1;\n  3:   108,    216,    171,     67,     13,      1;\n  4:   216,    540,    558,    305,     93,     15,     1;\n  5:   648,   1836,   2214,   1473,    584,    138,    18,     1;\n  6:  1296,   4320,   6264,   5160,   2641,    860,   174,    20,    1;\n  7:  3888,  14256,  23112,  21744,  13083,   5221,  1382,   234,   23,   1;\n  8:  7776,  32400,  60480,  66600,  47910,  23525,  7985,  1850,  280,  25,  1;\n  9: 23328, 104976, 213840, 260280, 210330, 118485, 47480, 13535, 2690, 355, 28, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(5*n+2+(2*n+1)*x)*P(n-1,x) + (12+10*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6 + 5*x + x^2.","mathematica":"P[0, x_] := 6 + 5*x + x^2;\nP[n_, x_] := P[n, x] = Expand[-(5*n + 2 + (2*n + 1)*x)*P[n - 1, x] + (12 + 10*x + 2*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 6 + 5*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = -(5*n + 2 + (2*n + 1)*x)*v[i - 1] + (12 + 10*x + 2*x^2)*deriv(v[i - 1], x)); 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-5a70a6e97131d437","steps":["reverse","x \u21a6 x+1"],"edges":["e-75e40465d2024023","e-f87cc16a200c95f5"],"text":"A064861 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A064861 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}